# questions

title: the-greatest-and-least-values-of-sin-1-x-2-cos-1-x-2-are-respectively
slug: the-greatest-and-least-values-of-sin-1-x-2-cos-1-x-2-are-respectively-62113
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P2.db
question:
<div class="question">The greatest and least values of $\left(\sin ^{-1} x\right)^{2}+\left(\cos ^{-1} x\right)^{2}$ are respectively</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\pi^{2}}{4}$ and 0</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\pi}{2}$ and $\frac{-\pi}{2}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{5 \pi^{2}}{4}$ and $\frac{\pi^{2}}{8}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\pi^{2}}{4}$ and $\frac{-\pi^{2}}{4}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{5 \pi^{2}}{4}$ and $\frac{\pi^{2}}{8}$</span> </div>

answer:
<div class="solution">We have, $\left(\sin ^{-1} x\right)^{2}+\left(\cos ^{-1} x\right)^{2}$
<br/> <br/>$=\left(\sin ^{-1} x+\cos ^{-1} x\right)^{2}-2 \sin ^{-1} x \cdot \cos ^{-1} x$
<br/> <br/>$=\frac{\pi^{2}}{4}-2 \sin ^{-1} x\left(\frac{\pi}{2}-\sin ^{-1} x\right)$
<br/> <br/>$=\frac{\pi^{2}}{4}-\pi \sin ^{-1} x+2\left(\sin ^{-1} x\right)^{2}$
<br/> <br/>$=2\left[\left(\sin ^{-1} x\right)^{2}-\frac{\pi}{2} \sin ^{-1} x+\frac{\pi^{2}}{8}\right]$
<br/> <br/>$=2\left[\left(\sin ^{-1} x-\frac{\pi}{4}\right)^{2}+\frac{\pi^{2}}{16}\right]$
<br/> <br/>Thus, the least value is $2\left(\frac{\pi^{2}}{16}\right)$ i.e. $\frac{\pi^{2}}{8}$
<br/> <br/>and the greatest value is $2\left[\left(\frac{-\pi}{2}-\frac{\pi}{4}\right)^{2}+\frac{\pi^{2}}{16}\right]$ i.e. $\frac{5 \pi^{2}}{4} .$</div>


title: the-greatest-positive-integer-which-divides-n-n-1-n-2-n-3-for-all-n-n-is
slug: the-greatest-positive-integer-which-divides-n-n-1-n-2-n-3-for-all-n-n-is-58175
tags: ['Mathematics', 'Basic of Mathematics', 'BITSAT', 'BITSAT 2014']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">24</span> </div>
source_db: MarksBatch2_P2.db
question:
<div class="question">The greatest positive integer, which divides
<br/> <br/>$\mathrm{n}(\mathrm{n}+1)(\mathrm{n}+2)(\mathrm{n}+3)$ for all $\mathrm{n} \in \mathbf{N},$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">6</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">24</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">120</span> </li> </ul>

answer:
<div class="solution">The product of $\mathrm{r}$ consecutive integers is divisible byr! . Thus $n(n+1)(n+2)(n+3)$ is divisible by $4 !=24$</div>


title: the-greatest-term-in-the-expansion-of-3-1-1-3-20-is
slug: the-greatest-term-in-the-expansion-of-3-1-1-3-20-is-28186
tags: ['Mathematics', 'Binomial Theorem', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P2.db
question:
<div class="question">The greatest term in the expansion of <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>3</mn></msqrt><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac></mrow></mfenced><mn>20</mn></msup></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>26840</mn><mn>9</mn></mfrac></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>24840</mn><mn>9</mn></mfrac></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>25840</mn><mn>9</mn></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>25840</mn><mn>9</mn></mfrac></math></span> </div>

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msqrt><mn>3</mn></msqrt><msup><mo>·</mo><mn>20</mn></msup><msub><mi>C</mi><mi>r</mi></msub><msup><mfenced><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac></mfenced><mi>r</mi></msup></math></p><p>and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mi>r</mi></msub><mo>=</mo><msqrt><mn>3</mn></msqrt><msup><mo>·</mo><mn>20</mn></msup><msub><mi>C</mi><mrow><mi>r</mi><mo>-</mo><mn>1</mn></mrow></msub><msup><mfenced><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac></mfenced><mrow><mi>r</mi><mo>-</mo><mn>1</mn></mrow></msup></math></p><p>Now, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>T</mi><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msub><msub><mi>T</mi><mi>r</mi></msub></mfrac><mo>=</mo><mfrac><mrow><mn>20</mn><mo>-</mo><mi>r</mi><mo>+</mo><mn>1</mn></mrow><mi>r</mi></mfrac><mfenced><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac></mfenced></math></p><p>Since, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mrow><mi>r</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>≥</mo><msub><mi>T</mi><mi>r</mi></msub></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>20</mn><mo>-</mo><mi>r</mi><mo>+</mo><mn>1</mn><mo>≥</mo><msqrt><mn>3</mn></msqrt><mi>r</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>r</mi><mo>≤</mo><mfrac><mn>21</mn><mrow><msqrt><mn>3</mn></msqrt><mo>+</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mfrac><mn>21</mn><mrow><mn>2</mn><mo>.</mo><mn>73</mn></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>r</mi><mo>≤</mo><mn>7</mn><mo>.</mo><mn>692</mn><mo>⇒</mo><mi>r</mi><mo>=</mo><mn>7</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> The greatest term is <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>8</mn></msub><mo>=</mo><msqrt><mn>3</mn></msqrt><mo>·</mo><mmultiscripts><mi>C</mi><mn>7</mn><none></none><mprescripts></mprescripts><none></none><mn>20</mn></mmultiscripts><msup><mfenced><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac></mfenced><mn>7</mn></msup><mo>=</mo><mfrac><mn>25840</mn><mn>9</mn></mfrac></math></p></div>


title: the-greenhouse-effect-is-because-of-the-1
slug: the-greenhouse-effect-is-because-of-the-1-91955
question: <div class="question">The greenhouse effect is because of the</div>
tags: ['Chemistry', 'Environmental Chemistry', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">presence of gases, which in general are strong infrared absorbers, in the atmosphere</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">presence of $\mathrm{CO}_{2}$, only in the atmosphere</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">pressure of $\mathrm{O}_{3}$ and $\mathrm{CH}_{4}$ in the atmosphere</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{N}_{2} \mathrm{O}$ and chlorofluorohydrocarbons in the atmosphere</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">presence of gases, which in general are strong infrared absorbers, in the atmosphere</span> </div>

answer:
<div class="solution">Green house gases such as $\mathrm{CO}_{2}$, ozone, methane, the chlorofluorocarbon compounds and water vapour form a thick cover around the earth which prevents the IR rays emitted by the earth to escape. It gradually leads to increase in temperature of atmosphere.</div>


title: the-greenhouse-effect-is-because-of-the
slug: the-greenhouse-effect-is-because-of-the-34654
question: <div class="question">The greenhouse effect is because of the</div>
tags: ['Chemistry', 'Environmental Chemistry', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">presence of gases, which in general are strong infrared absorbers, in the atmosphere.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">presence of $\mathrm{CO}_{2}$, only in the atmosphere.</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">presence of $\mathrm{O}_{3}$ and $\mathrm{CH}_{4}$ in the atmosphere.</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{N}_{2} \mathrm{O}$ and chlorofluorohydrocarbons in the atmosphere.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">presence of gases, which in general are strong infrared absorbers, in the atmosphere.</span> </div>

answer:
<div class="solution">Green house gases such as $\mathrm{CO}_{2}$, ozone, methane, the chlorofluoro carbon compounds and water vapour form a thick cover around the earth which prevents the IR rays emitted by the earth to escape. It gradually leads to increase in temperature of atmosphere.</div>


tags: ['Physics', 'Mathematics in Physics', 'BITSAT', 'BITSAT 2020']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$6 \%$</span> </div>
source_db: MarksBatch2_P2.db
title:
the-heat-generated-in-a-circuit-is-given-by-q-i-2-rt-where-i-is-current-r-is-resistance-and-tis-time-if-the-percentage-errors-in-measuring-i-r-and-t-a

slug:
the-heat-generated-in-a-circuit-is-given-by-q-i-2-rt-where-i-is-current-r-is-resistance-and-tis-time-if-the-percentage-errors-in-measuring-i-r-and-t-a-32699

question:
<div class="question">The heat generated in a circuit is given by $\mathrm{Q}=\mathrm{I}^{2} \mathrm{Rt},$ where I is current, $\mathrm{R}$ is resistance and tis time. If the percentage errors in measuring I. $\mathrm{R}$ and $\mathrm{t}$ are $2 \%, 1 \%$ and $1 \%$ respectively, then the maximum error in measuring heat will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2 \%$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$3 \%$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$4 \%$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$6 \%$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">$\begin{aligned} \frac{\Delta Q}{Q} \times 100 &amp;=\frac{2 \Delta I}{I} \times 100+\frac{\Delta R}{R} \times 100+\frac{\Delta t}{t} \times 100 \\ &amp;=2 \times 2 \%+1 \%+1 \%=6 \% \end{aligned}$</div>


tags: ['Chemistry', 'Thermodynamics (C)', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P2.db
title:
the-heat-of-reaction-for-c-10-h-8-s-12-o-2-g-10-co-2-g-4-h-2-o-l-at-constant-volume-is-1228-2-kcal-at-25-0-c-the-heat-of-reaction-at-constant-pressure

slug:
the-heat-of-reaction-for-c-10-h-8-s-12-o-2-g-10-co-2-g-4-h-2-o-l-at-constant-volume-is-1228-2-kcal-at-25-0-c-the-heat-of-reaction-at-constant-pressure-64285

question:
<div class="question"><p>The heat of reaction for</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">C</mi><mn>10</mn></msub><msub><mi mathvariant="normal">H</mi><mn>8</mn></msub><mfenced><mi mathvariant="normal">s</mi></mfenced><mo>+</mo><mn>12</mn><msub><mi mathvariant="normal">O</mi><mn>2</mn></msub><mfenced><mi mathvariant="normal">g</mi></mfenced><mo>⟶</mo><mn>10</mn><msub><mi>CO</mi><mn>2</mn></msub><mfenced><mi mathvariant="normal">g</mi></mfenced><mo>+</mo><mn>4</mn><msub><mi mathvariant="normal">H</mi><mn>2</mn></msub><mi mathvariant="normal">O</mi><mo>(</mo><mi mathvariant="normal">l</mi><mo>)</mo></math> at constant volume is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1228</mn><mo>.</mo><mn>2</mn><mo> </mo><mi>kcal</mi></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>25</mn><mn>0</mn></msup><mi mathvariant="normal">C</mi></math>. The heat of reaction at constant pressure and same temperature is</p></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1228</mn><mo>.</mo><mn>2</mn><mo> </mo><mi>kcal</mi></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1229</mn><mo>.</mo><mn>3</mn><mo> </mo><mi>kcal</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1232</mn><mo>.</mo><mn>9</mn><mo> </mo><mi>kcal</mi></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1242</mn><mo>.</mo><mn>6</mn><mo> </mo><mi>kcal</mi></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1229</mn><mo>.</mo><mn>3</mn><mo> </mo><mi>kcal</mi></math></span> </div>

answer:
<div class="solution"><blockquote><p>Heat of reaction at constant pressure is represented by ΔH</p><p>Heat of reaction at constant volume is represented by ΔU</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">C</mi><mn>10</mn></msub><msub><mi mathvariant="normal">H</mi><mn>8</mn></msub><mfenced><mi mathvariant="normal">s</mi></mfenced><mo>+</mo><mn>12</mn><msub><mi mathvariant="normal">O</mi><mn>2</mn></msub><mfenced><mi mathvariant="normal">g</mi></mfenced><mo>⟶</mo><mn>10</mn><msub><mi>CO</mi><mn>2</mn></msub><mfenced><mi mathvariant="normal">g</mi></mfenced><mo>+</mo><mn>4</mn><msub><mi mathvariant="normal">H</mi><mn>2</mn></msub><mi mathvariant="normal">O</mi></math></p><p>We known that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ΔH</mi><mo>=</mo><mi>ΔE</mi><mo>+</mo><msub><mi>Δn</mi><mi mathvariant="normal">g</mi></msub><mi>RT</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Δn</mi><mi>g</mi></msub><mo>=</mo><mn>10</mn><mo>-</mo><mn>12</mn><mo>=</mo><mo>-</mo><mn>2</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ΔH</mi><mo>=</mo><mo>-</mo><mn>1228</mn><mo>.</mo><mn>2</mn><mo>×</mo><msup><mn>10</mn><mn>3</mn></msup><mo>+</mo><mfenced><mrow><mo>-</mo><mn>2</mn></mrow></mfenced><mo>×</mo><mn>2</mn><mo>×</mo><mn>298</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mn>1229393</mn><mo> </mo><mi>Cal</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mn>1229</mn><mo>.</mo><mn>39</mn><mo> </mo><mi>kcal</mi></math></p></blockquote></div>


tags: ['Chemistry', 'Thermodynamics (C)', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P2.db
title:
the-heats-of-neutralisation-of-ch-3-cooh-hcooh-hcn-and-h-2-s-are-13213429-and-38-kcal-per-equivalent-respectively-arrange-the-acids-in-increasing-orde

slug:
the-heats-of-neutralisation-of-ch-3-cooh-hcooh-hcn-and-h-2-s-are-13213429-and-38-kcal-per-equivalent-respectively-arrange-the-acids-in-increasing-orde-38731

question:
<div class="question">The heats of neutralisation of $\mathrm{CH}_{3} \mathrm{COOH}$, $\mathrm{HCOOH}, \mathrm{HCN}$ and $\mathrm{H}_{2} \mathrm{~S}$ are -13.2,-13.4,-2.9
<br/> <br/>and $-3.8 \mathrm{kCal}$ per equivalent respectively. Arrange the acids in increasing order of acidic strength.</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\mathrm{HCOOH}&gt;\mathrm{CH}_{3} \mathrm{COOH}&gt;\mathrm{H}_{2} \mathrm{~S}&gt;\mathrm{HCN}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{CH}_{3} \mathrm{COOH}&gt;\mathrm{HCOOH}&gt;\mathrm{H}_{2} \mathrm{~S}&gt;\mathrm{HCN}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{H}_{2} \mathrm{~S}&gt;\mathrm{HCOOH}&gt;\mathrm{CH}_{3} \mathrm{COOH}&gt;\mathrm{HCN}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{HCOOH}&gt;\mathrm{H}_{2} \mathrm{~S}&gt;\mathrm{CH}_{3} \mathrm{COOH}&gt;\mathrm{HCN}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{HCOOH}&gt;\mathrm{CH}_{3} \mathrm{COOH}&gt;\mathrm{H}_{2} \mathrm{~S}&gt;\mathrm{HCN}$</span> </div>

answer:
<div class="solution">The greater the (negative value) of heat of neutralisation, the more is the strength of the acid. Hence,
<br/> <br/>$$
<br/> <br/>\mathrm{HCOOH}&gt;\mathrm{CH}_{3} \mathrm{COOH}&gt;\mathrm{H}_{2} \mathrm{~S}&gt;\mathrm{HCN}
<br/> <br/>$$</div>


title: the-helical-structure-of-protein-is-stabilised-by
slug: the-helical-structure-of-protein-is-stabilised-by-30919
question: <div class="question">The helical structure of protein is stabilised by</div>
tags: ['Chemistry', 'Biomolecules', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">peptide bonds</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">dipeptide bonds</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">hydrogen bonds</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">vander waals forces</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">hydrogen bonds</span> </div>

answer:
<div class="solution">Fibrous proteins have thread like molecules which lie side by side to form fibres. The various molecules are held together by hydrogen bonds.</div>


title: the-helical-structure-of-protein-is-stabilized-by-1
slug: the-helical-structure-of-protein-is-stabilized-by-1-60869
question: <div class="question">The helical structure of protein is stabilized by</div>
tags: ['Chemistry', 'Biomolecules', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">dipeptide bonds</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">hydrogen bonds</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">ether bonds</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">peptide bonds</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">hydrogen bonds</span> </div>

answer:
<div class="solution">The $\alpha$-helix structure is formed when the chain of $\alpha$-amino acids coils as a right handed screw (called $\alpha$-helix) because of the formation of hydrogen bonds between amide groups of the same peptide chain, i.e., NH group in one unit is linked to carbonyl oxygen of the third unit by hydrogen bonding. This hydrogen bonding between different units is responsible for holding helix in a position.</div>


tags: ['Physics', 'Motion In Two Dimensions', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P2.db
title:
the-horizontal-range-and-maximum-height-attained-by-a-projectile-are-r-and-h-respectively-if-a-constant-horizontal-acceleration-a-g-4-is-imparted-to-t

slug:
the-horizontal-range-and-maximum-height-attained-by-a-projectile-are-r-and-h-respectively-if-a-constant-horizontal-acceleration-a-g-4-is-imparted-to-t-93823

question:
<div class="question">The horizontal range and maximum height attained by a projectile are <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi></math> respectively. If a constant horizontal acceleration <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mi>g</mi><mn>4</mn></mfrac></math> is imparted to the projectile due to wind, then its horizontal range and maximum height will be </div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>R</mi><mo>+</mo><mi>H</mi></mrow></mfenced><mo>,</mo><mfrac><mi>H</mi><mn>2</mn></mfrac></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>R</mi><mo>+</mo><mfrac><mi>H</mi><mn>2</mn></mfrac></mrow></mfenced><mo>,</mo><mn>2</mn><mi>H</mi></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>R</mi><mo>+</mo><mn>2</mn><mi>H</mi></mrow></mfenced><mo>,</mo><mi>H</mi></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>R</mi><mo>+</mo><mi>H</mi></mrow></mfenced><mo>,</mo><mi>H</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>R</mi><mo>+</mo><mi>H</mi></mrow></mfenced><mo>,</mo><mi>H</mi></math></span> </div>

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>=</mo><mfrac><mrow><mn>2</mn><msub><mi>U</mi><mi>y</mi></msub></mrow><mi>g</mi></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi><mo>=</mo><mfrac><msubsup><mi>U</mi><mi>y</mi><mn>2</mn></msubsup><mrow><mn>2</mn><mi>g</mi></mrow></mfrac></math></p><p>Here maximum height will be same as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi></math> because there is no acceleration change in vertical direction.</p><p>and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><msub><mi>U</mi><mi>x</mi></msub><mo>×</mo><mi>T</mi></math></p><p>When a horizontal acceleration is also given to the projectile <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>U</mi><mi>y</mi></msub><mo>,</mo><mi>T</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi></math> will remain unchanged while range will become</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>R</mi><mo>'</mo></msup><mo>=</mo><msub><mi>U</mi><mi>x</mi></msub><mo>×</mo><mi>T</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>a</mi><msup><mi>T</mi><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>R</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfrac><mi>g</mi><mn>4</mn></mfrac><mfenced><mfrac><mrow><mn>4</mn><msubsup><mi>U</mi><mi>y</mi><mn>2</mn></msubsup></mrow><msup><mi>g</mi><mn>2</mn></msup></mfrac></mfenced><mo>=</mo><mi>R</mi><mo>+</mo><mi>H</mi><mo>  </mo><mfenced close="]" open="["><mrow><mo>∵</mo><mi>a</mi><mo>=</mo><mfrac><mi>g</mi><mn>4</mn></mfrac></mrow></mfenced></math></p><p>Therefore horizontal range &amp; maximum height will be <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>R</mi><mo>+</mo><mi>H</mi></mrow></mfenced><mo> </mo><mo>&amp;</mo><mo> </mo><mi>H</mi></math>.</p></div>


title: the-hybridisation-of-fe-is-k-3-fe-cn-6-is
slug: the-hybridisation-of-fe-is-k-3-fe-cn-6-is-57890
tags: ['Chemistry', 'Coordination Compounds', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P2.db
question:
<div class="question">The hybridisation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Fe</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">K</mi><mn>3</mn></msub><mo>[</mo><mi>Fe</mi><mo>(</mo><mi>CN</mi><msub><mo>)</mo><mn>6</mn></msub><mo>]</mo></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>sp</mi><mn>3</mn></msup></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>dsp</mi><mn>3</mn></msup></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>sp</mi><mn>3</mn></msup><msup><mi mathvariant="normal">d</mi><mn>2</mn></msup></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="normal">d</mi><mn>2</mn></msup><msup><mi>sp</mi><mn>3</mn></msup></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="normal">d</mi><mn>2</mn></msup><msup><mi>sp</mi><mn>3</mn></msup></math></span> </div>

answer:
<div class="solution"><p>Electronic configuration of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Fe</mi><mo>=</mo><mo>[</mo><mi>Ar</mi><mo>]</mo><mn>4</mn><mo> </mo><msup><mi mathvariant="normal">s</mi><mn>2</mn></msup><mn>3</mn><mo> </mo><msup><mi mathvariant="normal">d</mi><mn>6</mn></msup></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>Fe</mi><mrow><mn>2</mn><mo>+</mo></mrow></msup><mo>=</mo><mo>[</mo><mi>Ar</mi><mo>]</mo><mn>3</mn><msup><mi mathvariant="normal">d</mi><mn>6</mn></msup></math></p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/e4b105c2-28c7-4742-ac7e-a200f29b22d6-image.png" style="width: 400px; height: 78px;"/></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>CN</mi><mo>-</mo></msup></math> are strong field ligands</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/6019f557-2bcb-415a-bcc6-8620cc2a7d96-image.png" style="width: 400px; height: 58px;"/></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="normal">d</mi><mn>2</mn></msup><msup><mi>sp</mi><mn>3</mn></msup></math> hybridisation</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∵</mo><mo> </mo><msup><mi>CN</mi><mo>-</mo></msup></math> is a strong ligand and from six coordinate bonds with <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>Fe</mi><mrow><mn>3</mn><mo>+</mo></mrow></msup></math> . Thus <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Fe</mi></math> show <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="normal">d</mi><mn>2</mn></msup><msup><mi>sp</mi><mn>3</mn></msup></math> hybridisation in <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">K</mi><mn>3</mn></msub><mi>Fe</mi><mo>(</mo><mi>CN</mi><msub><mo>)</mo><mn>6</mn></msub></math>.</p></div>


title: the-hybridisation-of-phosphorous-in-po-4-3-is-1
slug: the-hybridisation-of-phosphorous-in-po-4-3-is-1-94215
tags: ['Chemistry', 'p Block Elements (Group 15, 16, 17 & 18)', 'JEE Main']
source_db: MarksBatch2_P2.db
question:
<div class="question">The hybridisation of phosphorous in <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>PO</mi><mn>4</mn><mrow><mn>3</mn><mo>-</mo></mrow></msubsup></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sp</mi></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>sp</mi><mn>2</mn></msup></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>sp</mi><mn>3</mn></msup></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>sp</mi><mn>3</mn></msup><mi mathvariant="normal">d</mi></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>sp</mi><mn>3</mn></msup></math></span> </div>

answer:
<div class="solution"><p>The structure of <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>PO</mi><mn>4</mn><mrow><mn>3</mn><mo>-</mo></mrow></msubsup></math> is as follows</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/d43adad2-02bd-4518-820f-6dba48b24acd-image.png" style="width: 222px; height: 252px;"/></p><p>In the given molecule, The P atom have 5 valence electrons. The hybridization can be determined by the number of sigma bonds and the number of lone pair.</p><p>As phosphorous central atom forms four <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mn>4</mn></mfenced></math> sigma bonds and has no lone pair of electrons. It uses four hybrid orbital and thus, hybridisation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">p</mi></math>-atom is <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>sp</mi><mn>3</mn></msup></math>.</p></div>


title: the-hypothetical-complex-chlorodiaquatriamminecobalt-iii-chloride-can-be-represented-as
slug: the-hypothetical-complex-chlorodiaquatriamminecobalt-iii-chloride-can-be-represented-as-55206
tags: ['Chemistry', 'Coordination Compounds', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch2_P2.db
question:
<div class="question">The hypothetical complex chlorodiaquatriamminecobalt (III) chloride can be represented as</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\left[\mathrm{CoCl}\left(\mathrm{NH}_{3}\right)_{3}\left(\mathrm{H}_{2} \mathrm{O}\right)_{2}\right] \mathrm{Cl}_{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{3}\left(\mathrm{H}_{2} \mathrm{O}\right) \mathrm{Cl}_{3}\right]$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{3}\left(\mathrm{H}_{2} \mathrm{O}\right)_{2} \mathrm{Cl}\right]$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{3}\left(\mathrm{H}_{2} \mathrm{O}\right)_{3}\right] \mathrm{Cl}_{3}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left[\mathrm{CoCl}\left(\mathrm{NH}_{3}\right)_{3}\left(\mathrm{H}_{2} \mathrm{O}\right)_{2}\right] \mathrm{Cl}_{2}$</span> </div>

answer:
<div class="solution">The complex chlorodiaquatriammine cobalt (III) chloride can have the structure $\left[\mathrm{CoCl}\left(\mathrm{NH}_{3}\right)_{3}\left(\mathrm{H}_{2} \mathrm{O}\right)_{2}\right] \mathrm{Cl}_{2}$</div>


title: the-i-v-characteristics-shown-in-figure-represents
slug: the-i-v-characteristics-shown-in-figure-represents-67318
tags: ['Physics', 'Semiconductors', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P2.db
question:
<div class="question">The I - V characteristics shown in figure represents
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/vxpZWooNuwTasMOKuqUcuRVgec6R4SaNp8uIcDnqzXU.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">ohmic conductors</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">non-ohmic conductors</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">insulators</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">superconductors</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">non-ohmic conductors</span> </div>

answer:
<div class="solution">The figure is showing I - V characteristics of non ohmic or non linear conductors.</div>


title: the-image-of-the-point-1-6-3-in-the-line-x-1-y-1-2-z-2-3-is
slug: the-image-of-the-point-1-6-3-in-the-line-x-1-y-1-2-z-2-3-is-89382
tags: ['Mathematics', 'Three Dimensional Geometry', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P2.db
question:
<div class="question">The image of the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>3</mn><mo>)</mo></math> in the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>x</mi><mn>1</mn></mfrac><mo>=</mo><mfrac><mrow><mi>y</mi><mo>-</mo><mn>1</mn></mrow><mn>2</mn></mfrac><mo>=</mo><mfrac><mrow><mi>z</mi><mo>-</mo><mn>2</mn></mrow><mn>3</mn></mfrac></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>7</mn><mo>)</mo></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mo>-</mo><mn>7</mn><mo>)</mo></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>7</mn><mo>)</mo></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><span style="background-color:null;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>2</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>7</mn><mo>)</mo></math></span></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>7</mn><mo>)</mo></math></span> </div>

answer:
<div class="solution"><p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>3</mn><mo>)</mo></math> be the given point and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> be the foot of perpendicular from <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> to the given line. The coordinates of a general point on the given line are</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/86a34f73-b58a-4256-9a66-d25f58d4e491-image.png" style="width: 250px; height: 290px;"/></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>x</mi><mo>-</mo><mn>0</mn></mrow><mn>1</mn></mfrac><mo>=</mo><mfrac><mrow><mi>y</mi><mo>-</mo><mn>1</mn></mrow><mn>2</mn></mfrac><mo>=</mo><mfrac><mrow><mi>z</mi><mo>-</mo><mn>2</mn></mrow><mn>3</mn></mfrac><mo>=</mo><mi>λ</mi></math>   (say)</p><p>i.e. <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>λ</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>2</mn><mi>λ</mi><mo>+</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>z</mi><mo>=</mo><mn>3</mn><mi>λ</mi><mo>+</mo><mn>2</mn></math> </p><p>If the coordinates of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> are <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>λ</mi><mo>,</mo><mn>2</mn><mi>λ</mi><mo>+</mo><mn>1</mn><mo>,</mo><mn>3</mn><mi>λ</mi><mo>+</mo><mn>2</mn><mo>)</mo></math>, then the direction ratios of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>L</mi></math> are <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>λ</mi><mo>-</mo><mn>1</mn><mo>,</mo><mn>2</mn><mi>λ</mi><mo>-</mo><mn>5</mn><mo>,</mo><mn>3</mn><mi>λ</mi><mo>-</mo><mn>1</mn><mo>)</mo></math>.</p><p>Since, the direction ratios of given line which is perpendicular to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>L</mi></math>, are <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>,</mo><mn>2</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math>. Therefore, <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>λ</mi><mo>-</mo><mn>1</mn><mo>)</mo><mn>1</mn><mo>+</mo><mo>(</mo><mn>2</mn><mi>λ</mi><mo>-</mo><mn>5</mn><mo>)</mo><mn>2</mn><mo>+</mo><mo>(</mo><mn>3</mn><mi>λ</mi><mo>-</mo><mn>1</mn><mo>)</mo><mn>3</mn><mo>=</mo><mn>0</mn></math>,which gives <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mn>1</mn></math>.</p><p>Hence, coordinates of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> are <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>5</mn><mo>)</mo></math>.</p><p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mfenced><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><msub><mi>y</mi><mn>1</mn></msub><mo>,</mo><msub><mi>z</mi><mn>1</mn></msub></mrow></mfenced></math> be the image of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>3</mn><mo>)</mo></math> in the given line. Then, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> is the mid-point of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>Q</mi></math>.</p><p>Therefore, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><mn>1</mn></mrow><mn>2</mn></mfrac><mo>=</mo><mn>1</mn><mo>,</mo><mfrac><mrow><msub><mi>y</mi><mn>1</mn></msub><mo>+</mo><mn>6</mn></mrow><mn>2</mn></mfrac><mo>=</mo><mn>3</mn><mo>,</mo><mfrac><mrow><msub><mi>z</mi><mn>1</mn></msub><mo>+</mo><mn>3</mn></mrow><mn>2</mn></mfrac><mo>=</mo><mn>5</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msub><mi>x</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>y</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>z</mi><mo>=</mo><mn>7</mn></math></p><p>Hence, the image of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mn>6</mn><mo>,</mo><mn>3</mn><mo>)</mo></math> in the given line is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>7</mn><mo>)</mo></math>.</p></div>


tags: ['Physics', 'Alternating Current', 'BITSAT', 'BITSAT 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0.6</span> </div>
source_db: MarksBatch2_P2.db
title:
the-impedance-of-a-circuit-consists-of-3-resistance-and-4-reactance-the-power-factor-of-the-circuit-is

slug:
the-impedance-of-a-circuit-consists-of-3-resistance-and-4-reactance-the-power-factor-of-the-circuit-is-39407

question:
<div class="question">The impedance of a circuit consists of $3 \Omega$ resistance and $4 \Omega$ reactance. The power factor of the circuit is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0.4</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">0.6</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">0.8</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1</span> </li> </ul>

answer:
<div class="solution">$\tan \phi=\frac{\mathrm{X}}{\mathrm{R}} \times \frac{4}{3}$
<br/>Power factor $=\cos \phi=\frac{3}{5}=0.6$</div>


title: the-important-antioxidant-used-in-food-is
slug: the-important-antioxidant-used-in-food-is-74522
question: <div class="question">The important antioxidant used in food is</div>
tags: ['Chemistry', 'Chemistry in Everyday Life', 'BITSAT', 'BITSAT 2016']
answer: <div class="solution">BHT is the important antioxidant used in food.</div>
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\mathrm{BHT}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{BHC}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{BTX}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">All the three</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{BHT}$</span> </div>


title: the-incorrect-statement-about-carbonate-co-3-2-ion-is
slug: the-incorrect-statement-about-carbonate-co-3-2-ion-is-19181
tags: ['Chemistry', 'Chemical Bonding and Molecular Structure', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P2.db
question:
<div class="question">The incorrect statement about carbonate <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><msubsup><mi>CO</mi><mn>3</mn><mrow><mn>2</mn><mo>-</mo></mrow></msubsup></mfenced></math> ion is,</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">It has planar structure</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">It has one coordinate bond</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">It has three resonating structure</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Hydrolysis of <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>CO</mi><mn>3</mn><mrow><mn>2</mn><mo>-</mo></mrow></msubsup></math> ion gives basic solution</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">It has one coordinate bond</span> </div>

answer:
<div class="solution"><p>Resonating structure of carbonate <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>CO</mi><mn>3</mn><mrow><mn>2</mn><mo>-</mo></mrow></msubsup></math> ion are</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/296d8d16-fa2d-456c-8c26-2b39438519d1-be6ca139-a28a-4b8f-96c4-d250dd3fe4c9-image.png" style="width: 955px; height: 185px;"/></p><p>In resonating structures bonds are not fixed and therefore, all bond lengths are equal.</p><p>The hybridisation of central atom is <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>sp</mi><mn>2</mn></msup></math>.</p><p>Thus, there is no coordinate bond present in <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>CO</mi><mn>3</mn><mrow><mn>2</mn><mo>-</mo></mrow></msubsup></math>. Hence, the second statement is incorrect.</p></div>


title: the-instantaneous-current-from-an-ac-source-is-i-6-sin-314-t-what-is-the-rms-value-of-the-current
tags: ['Physics', 'Alternating Current', 'BITSAT', 'BITSAT 2009']
source_db: MarksBatch2_P2.db
slug:
the-instantaneous-current-from-an-ac-source-is-i-6-sin-314-t-what-is-the-rms-value-of-the-current-14553

question:
<div class="question">The instantaneous current from an a.c. source is $\mathrm{I}=6 \sin 314 \mathrm{t}$. What is the rms value of the current?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$3 \sqrt{2} \mathrm{amp}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 \sqrt{2}$ amp</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\sqrt{2} \mathrm{amp}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2 \mathrm{amp}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$3 \sqrt{2} \mathrm{amp}$</span> </div>

answer:
<div class="solution">$\mathrm{I}_{\mathrm{rms}}=\frac{\mathrm{I}_{0}}{\sqrt{2}}=\frac{6}{\sqrt{2}}=3 \sqrt{2} \mathrm{amp} .$</div>


title: the-integral-1-x-1-x-e-x-1-x-d-x-is-equal-to-1
slug: the-integral-1-x-1-x-e-x-1-x-d-x-is-equal-to-1-12436
tags: ['Mathematics', 'Indefinite Integration', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P2.db
question:
<div class="question">The integral <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi><mo>-</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></mfenced><msup><mi>e</mi><mrow><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></msup><mi>d</mi><mi>x</mi></math> is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><msup><mi>e</mi><mrow><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></msup><mo>+</mo><mi>C</mi></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><msup><mi>e</mi><mrow><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></msup><mo>+</mo><mi>C</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><msup><mi>e</mi><mrow><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></msup><mo>+</mo><mi>C</mi></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi>x</mi><msup><mi>e</mi><mrow><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></msup><mo>+</mo><mi>C</mi></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><msup><mi>e</mi><mrow><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></msup><mo>+</mo><mi>C</mi></math></span> </div>

answer:
<div class="solution"><p>We have, </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mo>=</mo><mo>∫</mo><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi><mo>-</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></mfenced><msup><mi>e</mi><mrow><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></msup><mi>d</mi><mi>x</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mo>=</mo><mo>∫</mo><msup><mi>e</mi><mrow><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></msup><mi>d</mi><mi>x</mi><mo>+</mo><mo>∫</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></mfenced><msup><mi>e</mi><mrow><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></msup><mi>d</mi><mi>x</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mo>=</mo><mo>∫</mo><msup><mi>e</mi><mrow><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></msup><mi>d</mi><mi>x</mi><mo>+</mo><mo>∫</mo><mi>x</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mn>1</mn><msup><mi>x</mi><mn>2</mn></msup></mfrac></mrow></mfenced><msup><mi>e</mi><mrow><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></msup><mi>d</mi><mi>x</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mo>=</mo><mo>∫</mo><msup><mi>e</mi><mrow><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></msup><mi>d</mi><mi>x</mi><mo>+</mo><mi>x</mi><msup><mi>e</mi><mrow><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></msup><mo>-</mo><mo>∫</mo><msup><mi>e</mi><mrow><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></msup><mi>d</mi><mi>x</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mo>=</mo><mi>x</mi><msup><mi>e</mi><mrow><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mrow></msup><mo>+</mo><mi>C</mi></math></p></div>


title: the-integrating-factor-of-d-x-x-d-y-y-x-4-3-x-is-1
slug: the-integrating-factor-of-d-x-x-d-y-y-x-4-3-x-is-1-61833
question: <div class="question">The integrating factor of $\frac{x d y}{d x}-y=x^4-3 x$ is</div>
tags: ['Mathematics', 'Differential Equations', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>x</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\log x$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{1}{x}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-\mathbf{X}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{x}$</span> </div>

answer:
<div class="solution"><br/>$\begin{aligned}<br/>&amp; \text { (c) Since } x \frac{d y}{d x}-y=x^4-3 x \\<br/>&amp; \therefore \frac{d y}{d x}-\frac{y}{x}=x^3-3<br/>\end{aligned}$<br/>Hence $I F=e^{\int P d x}=e^{-\int \frac{1}{x} d x}=e^{-\log x}=\frac{1}{x}$</div>


title: the-intermediate-formed-during-the-addition-of-hcl-to-propene-in-the-presence-of-peroxide-is
slug: the-intermediate-formed-during-the-addition-of-hcl-to-propene-in-the-presence-of-peroxide-is-13262
tags: ['Chemistry', 'General Organic Chemistry', 'BITSAT', 'BITSAT 2009']
source_db: MarksBatch2_P2.db
question:
<div class="question">The intermediate formed during the addition of $\mathrm{HCl}$ to propene in the presence of peroxide is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{CH}_{3} \mathrm{CHCH}_{2} \mathrm{CI}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\mathrm{CH}_{2} \mathrm{CHCH}_{3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CH}_{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{C} \mathrm{H}_{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{CH}_{2} \mathrm{CHCH}_{3}$</span> </div>

answer:
<div class="solution">The addition of HCl to propene proceeds by ionic mechanism and not by free radical mechanism. Hence it forms intermediate carbonium ion.</div>


title: the-interval-in-which-the-function-2-x-3-15-increases-less-rapidly-than-the-function-9-x-2-12-x-is
tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2014']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">(1,2)</span> </div>
source_db: MarksBatch2_P2.db
slug:
the-interval-in-which-the-function-2-x-3-15-increases-less-rapidly-than-the-function-9-x-2-12-x-is-96236

question:
<div class="question">The interval in which the function $2 \mathrm{x}^{3}+15$ increases less rapidly than the function $9 x^{2}-12 x,$ is $-$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$(-\infty, 1)$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">(1,2)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$(2, \infty)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

answer:
<div class="solution">Let $f(x)=2 x^{3}+15$ and $g(x)=9 x^{2}-12 x$ then $\mathrm{f}^{\prime}(\mathrm{x})=6 \mathrm{x}^{2} \forall \mathrm{x} \in \mathrm{R}$
<br/> <br/>$\therefore \mathrm{f}(\mathrm{x})$ is increasing function $\forall \mathrm{x} \in \mathrm{R}$
<br/> <br/>$$
<br/> <br/>\text { Also, } g^{\prime}(x)&gt;0 \Rightarrow 18 x-12&gt;0 \Rightarrow x&gt;\frac{2}{3}
<br/> <br/>$$
<br/> <br/>Thus, $\mathrm{f}(\mathrm{x})$ and $\mathrm{g}(\mathrm{x})$ both increases for $\mathrm{x}&gt;\frac{2}{3}$
<br/> <br/>$$
<br/> <br/>\text { Let } F(x)=f(x)-g(x), F^{\prime}(x)&lt;0
<br/> <br/>$$
<br/> <br/>$(\because \mathrm{f}(\mathrm{x})$ increases less rapidly than the function $\mathrm{g}(\mathrm{x}))$
<br/> <br/>$$
<br/> <br/>\Rightarrow 6 \mathrm{x}^{2}-18 \mathrm{x}+12&lt;0 \Rightarrow 1&lt;\mathrm{x}&lt;2
<br/> <br/>$$</div>


title: the-interval-in-which-the-function-f-x-x-4-x-2-1-is-decreasing-is-1
slug: the-interval-in-which-the-function-f-x-x-4-x-2-1-is-decreasing-is-1-33881
tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
source_db: MarksBatch2_P2.db
question:
<div class="question">The interval in which the function $f(x)=$ $\frac{4 x^2+1}{x}$ is decreasing is :</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\left(-\frac{1}{2}, \frac{1}{2}\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\left[-\frac{1}{2}, \frac{1}{2}\right]$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$(-1,1)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$[-1,1]$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(-\frac{1}{2}, \frac{1}{2}\right)$</span> </div>

answer:
<div class="solution">Given $f(x)=\frac{4 x^2+1}{x}$ Thus $f^{\prime}(x)=4-\frac{1}{x^2}$<br/>$\mathrm{f}(\mathrm{x})$ will be decreasing if $\mathrm{f}^{\prime}(\mathrm{x}) &lt; 0$<br/>Thus $4-\frac{1}{x^2} &lt; 0 \Rightarrow \frac{1}{x^2}&gt;4 \Rightarrow \frac{-1}{2} &lt; x &lt; \frac{1}{2}$<br/>Thus interval in which $f(x)$ is decreasing, is $\left(-\frac{1}{2}, \frac{1}{2}\right)$.</div>


title: the-inverse-of-the-function-f-x-log-a-x-x-2-1-where-a-0-a-1-is
slug: the-inverse-of-the-function-f-x-log-a-x-x-2-1-where-a-0-a-1-is-38792
tags: ['Mathematics', 'Functions', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P2.db
question:
<div class="question">The inverse of the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msub><mi>log</mi><mi>a</mi></msub><mfenced><mrow><mi>x</mi><mo>+</mo><msqrt><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></msqrt></mrow></mfenced></math> (where, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>&lt;</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>a</mi><mo>≠</mo><mn>1</mn></math>) is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><msup><mi>a</mi><mi>x</mi></msup><mo>-</mo><msup><mi>a</mi><mrow><mo>-</mo><mi>x</mi></mrow></msup></mrow></mfenced></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">not defined for all <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">defined for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>&gt;</mo><mn>0</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><msup><mi>a</mi><mi>x</mi></msup><mo>-</mo><msup><mi>a</mi><mrow><mo>-</mo><mi>x</mi></mrow></msup></mrow></mfenced></math></span> </div>

answer:
<div class="solution"><p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>y</mi></math>, then</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mi>y</mi></msup><mo>=</mo><mi>x</mi><mo>+</mo><msqrt><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></msqrt></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>a</mi><mrow><mo>-</mo><mi>y</mi></mrow></msup><mo>=</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>+</mo><msqrt><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></msqrt></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>a</mi><mrow><mo>-</mo><mi>y</mi></mrow></msup><mo>=</mo><mfrac><mrow><mi>x</mi><mo>-</mo><msqrt><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></msqrt></mrow><mrow><mo>-</mo><mn>1</mn></mrow></mfrac></math>  [rationalising]</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><msup><mi>a</mi><mi>y</mi></msup><mo>-</mo><msup><mi>a</mi><mrow><mo>-</mo><mi>y</mi></mrow></msup><mo>=</mo><mn>2</mn><mi>x</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><msup><mi>a</mi><mi>y</mi></msup><mo>-</mo><msup><mi>a</mi><mrow><mo>-</mo><mi>y</mi></mrow></msup></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>y</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><msup><mi>a</mi><mi>y</mi></msup><mo>-</mo><msup><mi>a</mi><mrow><mo>-</mo><mi>y</mi></mrow></msup></mrow></mfenced></math>   <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><mo>∵</mo><mo> </mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>y</mi><mo>⇒</mo><mi>x</mi><mo>=</mo><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>y</mi></mfenced></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><msup><mi>a</mi><mi>x</mi></msup><mo>-</mo><msup><mi>a</mi><mrow><mo>-</mo><mi>x</mi></mrow></msup></mrow></mfenced></math>  [replacing <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>]</p><p>Hence, option (a) is correct.</p></div>


tags: ['Chemistry', 'Electrochemistry', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P2.db
title:
the-ionic-conductance-of-ba-2-and-cl-respectively-are-127-and-76-1-cm-2-at-infinite-dilution-the-equivalent-conductance-of-bacl-2-at-infinite-dilution

slug:
the-ionic-conductance-of-ba-2-and-cl-respectively-are-127-and-76-1-cm-2-at-infinite-dilution-the-equivalent-conductance-of-bacl-2-at-infinite-dilution-53031

question:
<div class="question">The ionic conductance of $\mathrm{Ba}^{2+}$ and $\mathrm{Cl}^{-}$ respectively are 127 and $76 \Omega^{-1} \mathrm{~cm}^{2}$ at infinite dilution. The equivalent conductance of $\mathrm{BaCl}_{2}$, at infinite dilution will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$330 \Omega^{-1} \mathrm{~cm}^{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$203 \Omega^{-1} \mathrm{~cm}^{2}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$139 \Omega^{-1} \mathrm{~cm}^{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$51 \Omega^{-1} \mathrm{~cm}^{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$139 \Omega^{-1} \mathrm{~cm}^{2}$</span> </div>

answer:
<div class="solution">$\begin{aligned} &amp; \lambda_{\mathrm{m}}^{\circ} \text { for } \mathrm{BaCl}_{2}=\lambda_{\mathrm{m}}^{\circ} \mathrm{Ba}^{2+}+\lambda_{\mathrm{m}}^{\circ} \mathrm{Cl}^{-} \\ &amp;=\frac{1}{2} \times 127+76=139.5 \Omega^{-1} \mathrm{~cm}^{2} \end{aligned}$</div>


title: the-ionization-constant-of-phenol-is-higher-than-that-of-ethanol-because
slug: the-ionization-constant-of-phenol-is-higher-than-that-of-ethanol-because-80991
tags: ['Chemistry', 'General Organic Chemistry', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P2.db
question:
<div class="question">The ionization constant of phenol is higher than that of ethanol because :</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Phenoxide ion is bulkier than ethoxide</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Phenoxide ion is stronger base than ethoxide</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Phenoxide ion is stabilized through delocalization</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Phenoxide ion is less stable than ethoxide</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Phenoxide ion is stabilized through delocalization</span> </div>

answer:
<div class="solution">The acidic nature of phenol is due to the formation of stable phenoxide ion in solution
<br/> <br/>$$
<br/> <br/>\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{OH}+\mathrm{H}_{2} \mathrm{O} \rightleftharpoons \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}_{\text {Phenoxide ion }}^{-}+\mathrm{H}_{3} \mathrm{O}^{+}
<br/> <br/>$$
<br/> <br/>The phenoxide ion is stable due to
<br/> <br/>resonance.
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/VttJ3r5W3I-CAaXjKp-AiYl2kqdRZoaZU0w-wqAHrK0.original.fullsize.png"/><br/> <br/> <br/>The negative charge is delocalized in the benzene ring which is a stabilizing factor in the phenoxide ion and increase acidity of phenol. wheras no resonance is possible in alkoxide ions $\left(\mathrm{RO}^{-}\right)$ derived from alcohol. The negative charge is localized on oxygen atom. Thus, alcohols are not acidic.</div>


title: the-ions-o-2-f-na-mg-2-and-al-3-are-isoelectronic-their-ionic-radii-show
slug: the-ions-o-2-f-na-mg-2-and-al-3-are-isoelectronic-their-ionic-radii-show-88891
tags: ['Chemistry', 'Classification of Elements and Periodicity in Properties', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P2.db
question:
<div class="question">The ions $\mathrm{O}^{2-}, \mathrm{F}^{-}, \mathrm{Na}^{+}, \mathrm{Mg}^{2+}$ and $\mathrm{Al}^{3+}$ are
<br/> <br/>isoelectronic. Their ionic radii show:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">A decrease from $\mathrm{O}^{2-}$ to $\mathrm{F}^{-}$ and then increase from $\mathrm{Na}^{+}$ to $\mathrm{Al}^{3+}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">A significant increase from $\mathrm{O}^{2-}$ to $\mathrm{Al}^{3+}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">A significant decrease from $\mathrm{O}^{2-}$ to $\mathrm{Al}^{3+}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">An increase from $\mathrm{O}^{2-}$ to $\mathrm{F}^{-}$ and then decrease from $\mathrm{Na}^{+}$ to $\mathrm{Al}^{3+}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">A significant decrease from $\mathrm{O}^{2-}$ to $\mathrm{Al}^{3+}$</span> </div>

answer:
<div class="solution">Amongst isoelectronic species, ionic radii of anion is more than that of cations. Further size of anion increase with increase in -ve charge and size of cation decrease with increase in $+$ ve charge. Hence ionic radii decreases from $\mathrm{O}^{-}$ to $\mathrm{Al}^{3+}$</div>


title: the-isoelectricpoint-of-a-colloidially-dispersed-material-is-the-ph-value-at-which
slug: the-isoelectricpoint-of-a-colloidially-dispersed-material-is-the-ph-value-at-which-40013
tags: ['Chemistry', 'Surface Chemistry', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P2.db
question:
<div class="question">The isoelectric-point of a colloidially dispersed material is the $\mathrm{pH}$ value at which</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">the dispersed phase migrate in an electric field.</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">the dispersed phase does not migrate in an electric field.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">the dispersed phase has $\mathrm{pH}$ equal to 7 .</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">the dispersed phase has pH equal to zero.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">the dispersed phase does not migrate in an electric field.</span> </div>

answer:
<div class="solution">At isoelectric point there is no migration of dispersed phase in an electric field.</div>


tags: ['Chemistry', 'Ionic Equilibrium', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P2.db
title:
the-k-sp-of-cus-ag-2-s-and-hgs-are-1-0-31-1-0-44-and-1-0-54-respectively-the-solubility-of-these-sulphides-are-in-the-order

slug:
the-k-sp-of-cus-ag-2-s-and-hgs-are-1-0-31-1-0-44-and-1-0-54-respectively-the-solubility-of-these-sulphides-are-in-the-order-37333

question:
<div class="question">The $\mathrm{K}_{\mathrm{sp}}$ of $\mathrm{CuS}, \mathrm{Ag}_{2} \mathrm{~S}$ and $\mathrm{HgS}$ are $10^{-31}, 10^{-44}$ and $10^{-54}$ respectively. The solubility of these sulphides are in the order :</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\quad \mathrm{Ag}_{2} \mathrm{~S}&gt;\mathrm{CuS}&gt;\mathrm{HgS}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\quad \mathrm{AgS}&gt;\mathrm{HgS}&gt;\mathrm{CuS}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{HgS}&gt;\mathrm{Ag}_{2} \mathrm{~S}&gt;\mathrm{CuS}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\quad\mathrm{CuS}&gt;\mathrm{Ag}_{2} \mathrm{~S}&gt;\mathrm{HgS}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\quad \mathrm{Ag}_{2} \mathrm{~S}&gt;\mathrm{CuS}&gt;\mathrm{HgS}$</span> </div>

answer:
<div class="solution">For CuS, solubility $=\left(10^{-31}\right)^{1 / 2}$;<br/>$$<br/>\begin{array}{l}<br/>\text { For } \mathrm{Ag}_{2} \mathrm{~S}=\left(\frac{\mathrm{K}_{\mathrm{sp}}}{4}\right)^{\frac{1}{3}}=\left(\frac{10^{-44}}{4}\right)^{\frac{1}{3}} \text { and } \\<br/>\text { for } \mathrm{HgS}=\left(10^{-54}\right)^{\frac{1}{2}}<br/>\end{array}<br/>$$</div>


tags: ['Mathematics', 'Straight Lines', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P2.db
title:
the-larger-of-two-angles-made-with-the-x-axis-of-a-straight-line-drawn-through-1-2-so-that-it-intersects-the-line-x-y-4-at-a-point-distant-6-3-from-th

slug:
the-larger-of-two-angles-made-with-the-x-axis-of-a-straight-line-drawn-through-1-2-so-that-it-intersects-the-line-x-y-4-at-a-point-distant-6-3-from-th-34486

question:
<div class="question">The larger of two angles made with the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math>-axis of a straight line drawn through <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></math> so that it intersects the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mn>4</mn></math> at a point distant <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>6</mn></msqrt><mo>/</mo><mn>3</mn></math> from the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>60</mn><mo>°</mo></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>75</mn><mo>°</mo></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>105</mn><mo>°</mo></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>75</mn><mo>°</mo></math></span> </div>

answer:
<div class="solution"><p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></math> equation of line is</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow><mrow><mi>cos</mi><mi>θ</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>y</mi><mo>-</mo><mn>2</mn></mrow><mrow><mi>sin</mi><mi>θ</mi></mrow></mfrac><mo>=</mo><mi>r</mi><mo>=</mo><mfrac><msqrt><mn>6</mn></msqrt><mn>3</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mi>x</mi><mo>=</mo><mfrac><msqrt><mn>6</mn></msqrt><mn>3</mn></mfrac><mi>cos</mi><mi>θ</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>y</mi><mo>=</mo><mfrac><msqrt><mn>6</mn></msqrt><mn>3</mn></mfrac><mi>sin</mi><mi>θ</mi><mo>+</mo><mn>2</mn></math></p><p>This point lies on line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mn>4</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mfrac><msqrt><mn>6</mn></msqrt><mn>3</mn></mfrac><mi>cos</mi><mi>θ</mi><mo>+</mo><mfrac><msqrt><mn>6</mn></msqrt><mn>3</mn></mfrac><mi>sin</mi><mi>θ</mi><mo>=</mo><mn>1</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>sin</mi><mi>θ</mi><mo>+</mo><mi>cos</mi><mi>θ</mi><mo>=</mo><mfrac><msqrt><mn>3</mn></msqrt><msqrt><mn>2</mn></msqrt></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>sin</mi><mfenced><mrow><mfrac><mi>π</mi><mn>4</mn></mfrac><mo>+</mo><mi>θ</mi></mrow></mfenced><mo>=</mo><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mfrac><mi>π</mi><mn>4</mn></mfrac><mo>+</mo><mi>θ</mi><mo>=</mo><mn>60</mn><mo>°</mo></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>120</mn><mo>°</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>θ</mi><mo>=</mo><mn>15</mn><mo>°</mo></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>75</mn><mo>°</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> Larger angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>75</mn><mo>°</mo></math></p></div>


title: the-lassaignes-extract-is-boiled-with-dil-hno-3-before-testing-for-halogens-because
slug: the-lassaignes-extract-is-boiled-with-dil-hno-3-before-testing-for-halogens-because-52710
tags: ['Chemistry', 'General Organic Chemistry', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch2_P2.db
question:
<div class="question">The Lassaigne's extract is boiled with dil. $\mathrm{HNO}_{3}$ before testing for halogens because</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">silver halides are soluble in $\mathrm{HNO}_{3}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\mathrm{Na}_{2} \mathrm{~S}$ and $\mathrm{NaCN}$ are decomposed by $\mathrm{HNO}_{3}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{Ag}_{2} \mathrm{~S}$ is soluble in $\mathrm{HNO}_{3}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{AgCN}$ is soluble in $\mathrm{HNO}_{3}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{Na}_{2} \mathrm{~S}$ and $\mathrm{NaCN}$ are decomposed by $\mathrm{HNO}_{3}$</span> </div>

answer:
<div class="solution">$\mathrm{Na}_{2} \mathrm{~S}$ and $\mathrm{NaCN},$ formed during fusion with metallic sodium, must be removed before adding $\mathrm{AgNO}_{3},$ otherwise black ppt. due to $\mathrm{Na}_{2} \mathrm{~S}$ or white precipitate due to $\mathrm{AgCN}$ will be formed and thus white precipitate of $\mathrm{AgCl}$ will not be identified easily.
<br/> <br/>$\mathrm{Na}_{2} \mathrm{~S}+2 \mathrm{AgNO}_{3} \longrightarrow 2 \mathrm{NaNO}_{3}+\underset{\text { Black }}{\mathrm{Ag}_{2} \mathrm{~S}} \downarrow$
<br/> <br/>$\mathrm{NaCN}+\mathrm{AgNO}_{3} \longrightarrow \mathrm{NaNO}_{3} + \underset{\text { White }}{\mathrm{Ag} \mathrm{CN}} \downarrow$
<br/> <br/>$\mathrm{NaCl}+\mathrm{AgNO}_{3} \longrightarrow \mathrm{NaNO}_{3}+\underset{\text { White }}{\mathrm{Ag} \mathrm{Cl}} \downarrow$
<br/> <br/>$\mathrm{Na}_{2} \mathrm{~S}+2 \mathrm{HNO}_{3} \stackrel{\text { boil }}{\longrightarrow} 2 \mathrm{NaNO}_{3}+\mathrm{H}_{2} \mathrm{~S} \uparrow$
<br/> <br/>$\mathrm{NaCN}+\mathrm{HNO}_{3} \stackrel{\text { boil }}{\longrightarrow} \mathrm{NaNO}_{3}+\mathrm{HCN} \uparrow$</div>


title: the-laws-of-electromagnetic-induction-have-been-used-in-the-construction-of-a
slug: the-laws-of-electromagnetic-induction-have-been-used-in-the-construction-of-a-82052
tags: ['Physics', 'Electromagnetic Induction', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P2.db
question:
<div class="question">The laws of electromagnetic induction have been used in the construction of a</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">galvanometer</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">voltmeter</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">electric motor</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">generator</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">generator</span> </div>

answer:
<div class="solution">Hint:
<br/>The laws of Electromagnetic Induction give us idea about the various ways in which we can produce Electric Current using Electric or Magnetic field.
<br/>Correct Option is D.
<br/>Explanation for correct answer:
<br/>- Galvanometer is a device which detects current in the circuit.
<br/>- Voltmeter is a device which measures potential difference between two points in the circuit.
<br/>- Electric motor is a device which converts Electrical Energy into Mechanical Energy.
<br/>- Electric Generator is an device which produces Electric Current from Mechanical energy, Chemical energy, or by using Magnetic or Electric Fields. The laws if Electromagnetic Induction are used in Electric Generator which produces Electric Current from Magnetic or Electric Fields.</div>


title: the-least-positive-nonintegral-solution-of-the-equation-sin-x-2-x-sin-x-2-is
slug: the-least-positive-nonintegral-solution-of-the-equation-sin-x-2-x-sin-x-2-is-31820
tags: ['Mathematics', 'Trigonometric Equations', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch2_P2.db
question:
<div class="question">The least positive non-integral solution of the equation $\sin \pi\left(x^{2}+x\right)=\sin \pi x^{2}$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">rational</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">irrational of the form $\sqrt{\mathrm{p}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">irrational of the form $\frac{\sqrt{\mathrm{p}}-1}{4},$ where $\mathrm{p}$ is an odd integer</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">irrational of the form $\frac{\sqrt{\mathrm{p}}+1}{4},$ where $\mathrm{p}$ is an even integer</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">rational</span> </div>

answer:
<div class="solution">We have, $\sin \pi\left(x^{2}+x\right)=\sin \pi x^{2}$
<br/> <br/>$\Rightarrow \quad \pi\left(\mathrm{x}^{2}+\mathrm{x}\right)=\mathrm{n} \pi+(-1)^{\mathrm{n}} \pi \mathrm{x}^{2}$
<br/> <br/>$\therefore$ Either $\mathrm{x}^{2}+\mathrm{x}=2 \mathrm{~m}+\mathrm{x}^{2} \Rightarrow \mathrm{x}=2 \mathrm{~m} \in \mathrm{I}$
<br/> <br/>or $\quad \mathrm{x}^{2}+\mathrm{x}=\mathrm{k}-\mathrm{x}^{2},$ where $\mathrm{k}$ is an odd integer
<br/> <br/>$\Rightarrow \quad 2 \mathrm{x}^{2}+\mathrm{x}-\mathrm{k}=0 \Rightarrow \mathrm{x}=\frac{-1 \pm \sqrt{1+8 \mathrm{k}}}{4}$
<br/> <br/>For least positive non-integral solution
<br/> <br/>is $x=\frac{1}{2},$ when $k=1$</div>


title: the-least-value-of-the-function-f-x-0-x-3-sin-x-4-cos-x-d-x-in-the-interval-5-4-4-3-is
slug: the-least-value-of-the-function-f-x-0-x-3-sin-x-4-cos-x-d-x-in-the-interval-5-4-4-3-is-22272
tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P2.db
question:
<div class="question"><p>The least value of the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msubsup><mo>∫</mo><mn>0</mn><mi>x</mi></msubsup><mfenced><mrow><mn>3</mn><mi>sin</mi><mi>x</mi><mo>+</mo><mn>4</mn><mi>cos</mi><mi>x</mi></mrow></mfenced><mi>d</mi><mi>x</mi></math> in the interval <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="]" open="["><mrow><mfrac><mrow><mn>5</mn><mi>π</mi></mrow><mn>4</mn></mfrac><mo>,</mo><mfrac><mrow><mn>4</mn><mi>π</mi></mrow><mn>3</mn></mfrac></mrow></mfenced></math> is</p></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>3</mn><mo>-</mo><msqrt><mn>3</mn></msqrt></mrow><mn>2</mn></mfrac></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>5</mn><mo>-</mo><mn>4</mn><msqrt><mn>3</mn></msqrt></mrow><mn>2</mn></mfrac></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>7</mn><mo>-</mo><mn>4</mn><msqrt><mn>3</mn></msqrt></mrow><mn>2</mn></mfrac></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>9</mn><mo>-</mo><mn>4</mn><msqrt><mn>3</mn></msqrt></mrow><mn>2</mn></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>9</mn><mo>-</mo><mn>4</mn><msqrt><mn>3</mn></msqrt></mrow><mn>2</mn></mfrac></math></span> </div>

answer:
<div class="solution"><p>We have, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msubsup><mo>∫</mo><mn>0</mn><mi>x</mi></msubsup><mfenced><mrow><mn>3</mn><mi>sin</mi><mi>x</mi><mo>+</mo><mn>4</mn><mi>cos</mi><mi>x</mi></mrow></mfenced><mi>d</mi><mi>x</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>3</mn><mi>sin</mi><mi>x</mi><mo>+</mo><mn>4</mn><mi>cos</mi><mi>x</mi><mo>=</mo><mn>5</mn><mi>sin</mi><mfenced><mrow><mi>x</mi><mo>+</mo><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mn>4</mn><mn>3</mn></mfrac></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>5</mn><mi>π</mi></mrow><mn>4</mn></mfrac><mo>&lt;</mo><mi>x</mi><mo>&lt;</mo><mfrac><mrow><mn>4</mn><mi>π</mi></mrow><mn>3</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>π</mi><mo>&lt;</mo><mi>x</mi><mo>+</mo><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mn>4</mn><mo>/</mo><mn>3</mn><mo>&lt;</mo><mn>2</mn><mi>π</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∵</mo><mo> </mo><msup><mi>f</mi><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>&lt;</mo><mn>0</mn></math>, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> is decreasing.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∵</mo></math> least value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mrow><mn>5</mn><mi>π</mi></mrow><mn>4</mn></mfrac><mo>,</mo><mfrac><mrow><mn>4</mn><mi>π</mi></mrow><mn>3</mn></mfrac></mrow></mfenced><mo>=</mo><mi>f</mi><mfenced><mfrac><mrow><mn>4</mn><mi>π</mi></mrow><mn>3</mn></mfrac></mfenced></math></p><p>least <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msubsup><mo>∫</mo><mn>0</mn><mrow><mn>4</mn><mi>π</mi><mo>/</mo><mn>3</mn></mrow></msubsup><mfenced><mrow><mn>3</mn><mi>sin</mi><mi>x</mi><mo>+</mo><mn>4</mn><mi>cos</mi><mi>x</mi></mrow></mfenced><mi>d</mi><mi>x</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msubsup><mfenced close="]" open="["><mrow><mo>-</mo><mn>3</mn><mi>cos</mi><mi>x</mi><mo>+</mo><mn>4</mn><mi>sin</mi><mi>x</mi></mrow></mfenced><mn>0</mn><mrow><mn>4</mn><mi>π</mi><mo>/</mo><mn>3</mn></mrow></msubsup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mn>3</mn><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mn>1</mn></mrow></mfenced><mo>+</mo><mn>4</mn><mfenced><mrow><mo>-</mo><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>9</mn><mo>-</mo><mn>4</mn><msqrt><mn>3</mn></msqrt></mrow><mn>2</mn></mfrac></math></p></div>


tags: ['Physics', 'Current Electricity', 'BITSAT', 'BITSAT 2009']
source_db: MarksBatch2_P2.db
title:
the-length-of-a-potentiometer-wire-is-a-cell-of-emf-e-is-balanced-at-a-length-3-from-the-positive-end-of-the-wire-if-the-length-of-the-wire-is-increas

slug:
the-length-of-a-potentiometer-wire-is-a-cell-of-emf-e-is-balanced-at-a-length-3-from-the-positive-end-of-the-wire-if-the-length-of-the-wire-is-increas-46645

question:
<div class="question">The length of a potentiometer wire is $\ell$. A cell of emf $E$ is balanced at a length $\ell / 3$ from the positive end of the wire. If the length of the wire is increased by $\ell / 2 .$ At what distance will be the same cell give a balance point.</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2 \ell / 3$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\ell / 2$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\quad \ell / 6$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$4 \ell / 3$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\ell / 2$</span> </div>

answer:
<div class="solution">Potential gradient in the first case $=\frac{\mathrm{E}_{0}}{\ell}$ $$<br/>\therefore \quad \mathrm{E}=\left(\frac{\ell}{3}\right) \cdot\left(\frac{\mathrm{E}_{0}}{\ell}\right)=\frac{\mathrm{E}_{0}}{3}<br/>$$<br/>Potential gradient in second case<br/>$$<br/>=\frac{E_{0}}{3 \ell / 2} \text { (x) } \frac{2 E_{0}}{3 \ell}<br/>$$<br/>From equations (i) and (ii),<br/>$$<br/>\frac{\mathrm{E}_{0}}{3}-\left(\frac{2 \mathrm{E}_{0}}{3 \ell}\right) \mathrm{x} \Rightarrow \mathrm{x}=\frac{\ell}{2}<br/>$$</div>


title: the-length-of-perpendicular-drawn-from-the-point-2-3-4-to-line-4-x-2-y-6-1-z-3-is
slug: the-length-of-perpendicular-drawn-from-the-point-2-3-4-to-line-4-x-2-y-6-1-z-3-is-52323
tags: ['Mathematics', 'Three Dimensional Geometry', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P2.db
question:
<div class="question">The length of perpendicular drawn from the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>)</mo></math> to line <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>4</mn><mo>-</mo><mi>x</mi></mrow><mn>2</mn></mfrac><mo>=</mo><mfrac><mi>y</mi><mn>6</mn></mfrac><mo>=</mo><mfrac><mrow><mn>1</mn><mo>-</mo><mi>z</mi></mrow><mn>3</mn></mfrac></math>, is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>3</mn><mn>7</mn></mfrac><msqrt><mn>101</mn></msqrt></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mn>7</mn></mfrac><msqrt><mn>101</mn></msqrt></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mn>7</mn></mfrac><msqrt><mn>103</mn></msqrt></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>3</mn><mn>7</mn></mfrac><msqrt><mn>103</mn></msqrt></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>3</mn><mn>7</mn></mfrac><msqrt><mn>101</mn></msqrt></math></span> </div>

answer:
<div class="solution"><p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> be the foot of the perpendicular drawn from <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>)</mo></math> to the given line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi></math>.</p><p>Then, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow><mrow><mo>-</mo><mn>2</mn></mrow></mfrac><mo>=</mo><mfrac><mi>y</mi><mn>6</mn></mfrac><mo>=</mo><mfrac><mrow><mi>z</mi><mo>-</mo><mn>1</mn></mrow><mrow><mo>-</mo><mn>3</mn></mrow></mfrac></math></p><p>Now, any point on the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi></math> is given by</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>4</mn><mo>-</mo><mn>2</mn><mi>λ</mi><mo>,</mo><mi>y</mi><mo>=</mo><mn>6</mn><mi>λ</mi><mo>,</mo><mi>z</mi><mo>=</mo><mn>1</mn><mo>-</mo><mn>3</mn><mi>λ</mi></math></p><p>The coordinates of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> are <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>4</mn><mo>-</mo><mn>2</mn><mi>λ</mi><mo>,</mo><mn>6</mn><mi>λ</mi><mo>,</mo><mn>1</mn><mo>-</mo><mn>3</mn><mi>λ</mi><mo>)</mo></math></p><p>The direction ratios of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>AP</mi></math> are</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>4</mn><mo>-</mo><mn>2</mn><mi>λ</mi><mo>-</mo><mn>2</mn><mo>,</mo><mn>6</mn><mi>λ</mi><mo>-</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo>-</mo><mn>3</mn><mi>λ</mi><mo>-</mo><mn>4</mn><mo>)</mo></math></p><p>i.e. <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>2</mn><mo>-</mo><mn>2</mn><mi>λ</mi><mo>,</mo><mn>6</mn><mi>λ</mi><mo>-</mo><mn>3</mn><mo>,</mo><mo>-</mo><mn>3</mn><mo>-</mo><mn>3</mn><mi>λ</mi><mo>)</mo></math></p><p>And the direction ratios of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi></math> are<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mo>,</mo><mn>6</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn></math>.</p><p>Given, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>P</mi><mo>⊥</mo><mi>l</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mo>-</mo><mn>2</mn><mo>(</mo><mn>2</mn><mo>-</mo><mn>2</mn><mi>λ</mi><mo>)</mo><mo>+</mo><mn>6</mn><mo>(</mo><mn>6</mn><mi>λ</mi><mo>-</mo><mn>3</mn><mo>)</mo><mo>-</mo><mn>3</mn><mo>(</mo><mo>-</mo><mn>3</mn><mo>-</mo><mn>3</mn><mi>λ</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>λ</mi><mo>=</mo><mfrac><mn>13</mn><mn>49</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mi>A</mi><msup><mi>P</mi><mn>2</mn></msup><mo>=</mo><mo>(</mo><mn>4</mn><mo>-</mo><mn>2</mn><mi>λ</mi><mo>-</mo><mn>2</mn><msup><mo>)</mo><mn>2</mn></msup><mo>+</mo><mo>(</mo><mn>6</mn><mi>λ</mi><mo>-</mo><mn>3</mn><msup><mo>)</mo><mn>2</mn></msup><mo>+</mo><mo>(</mo><mn>1</mn><mo>-</mo><mn>3</mn><mi>λ</mi><mo>-</mo><mn>4</mn><msup><mo>)</mo><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>22</mn><mo>-</mo><mn>26</mn><mi>λ</mi><mo>+</mo><mn>49</mn><msup><mi>λ</mi><mn>2</mn></msup></math></p><p>Put <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mfrac><mn>13</mn><mn>49</mn></mfrac></math>, we get</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><msup><mi>P</mi><mn>2</mn></msup><mo>=</mo><mfrac><mn>909</mn><mn>49</mn></mfrac><mo>⇒</mo><mi>A</mi><mi>P</mi><mo>=</mo><mfrac><mn>3</mn><mn>7</mn></mfrac><msqrt><mn>101</mn></msqrt></math></p></div>


title: the-length-of-the-chord-x-y-3-intercepted-by-the-circle-x-2-y-2-2-x-2-y-2-0-is
slug: the-length-of-the-chord-x-y-3-intercepted-by-the-circle-x-2-y-2-2-x-2-y-2-0-is-81476
tags: ['Mathematics', 'Circle', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch2_P2.db
question:
<div class="question">The length of the chord $x+y=3$ intercepted by the circle $x^{2}+y^{2}-2 x-2 y-2=0$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{7}{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{3 \sqrt{3}}{2}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\sqrt{14}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\sqrt{7}}{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\sqrt{14}$</span> </div>

answer:
<div class="solution">The centre of the circle is \(\mathrm{C}(1,1)\) and radius of the circle is 2 , perpendicular distance from \(\mathrm{C}\) on \(\mathrm{AB}\), the chord \(\mathrm{x}+\mathrm{y}=3\)
<br/> <br/>\(\begin{aligned} &amp;\mathrm{CD}=\left|\frac{1+1-3}{\sqrt{2}}\right|=\frac{1}{\sqrt{2}} \\ &amp;\Rightarrow \mathrm{AD}=\sqrt{4-\frac{1}{2}}=\sqrt{\frac{7}{2}}\left[\mathrm{AD}=\sqrt{\mathrm{AC}^{2}-\mathrm{CD}^{2}}\right] \end{aligned}\)
<br/> <br/>Hence, the length of the chord \(\mathrm{AB}=2 \mathrm{AD}=2 \sqrt{\frac{7}{2}}=\sqrt{14}\).
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/hse6532ghg73.png"/></div>


tags: ['Mathematics', 'Parabola', 'BITSAT', 'BITSAT 2020']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">10</span> </div>
source_db: MarksBatch2_P2.db
title:
the-length-of-the-latus-rectum-of-the-parabola-which-has-focus-at-11-and-the-directrix-is-4-x-3-y-24-0-is

slug:
the-length-of-the-latus-rectum-of-the-parabola-which-has-focus-at-11-and-the-directrix-is-4-x-3-y-24-0-is-39694

question:
<div class="question">The length of the latus rectum of the parabola which has focus at (-1,1) and the directrix is $4 \mathrm{x}$ $+3 y-24=0$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">4</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">6</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">8</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">10</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Length of latus rectum $=2 \times$ distance of focus from directrix $=2 \times \frac{|1-4+3-24|}{5}=10$</div>


tags: ['Mathematics', 'Parabola', 'JEE Main']
source_db: MarksBatch2_P2.db
title:
the-length-of-the-latusrectum-of-the-parabola-whose-focus-is-leftfracu22-g-sin-2-alphafracu22-g-cos-2-alpharight-and-directrix-is-yfracu22-g-is-1

slug:
the-length-of-the-latusrectum-of-the-parabola-whose-focus-is-leftfracu22-g-sin-2-alphafracu22-g-cos-2-alpharight-and-directrix-is-yfracu22-g-is-1-20264

question:
<div class="question">The length of the latus-rectum of the parabola whose focus is \(\left(\frac{u^2}{2 g} \sin 2 \alpha,-\frac{u^2}{2 g} \cos 2 \alpha\right)\) and directrix is \(y=\frac{u^2}{2 g}\), is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\frac{u^2}{g} \cos ^2 \alpha\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\frac{u^2}{g} \cos 2 \alpha\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(\frac{2 u^2}{g} \cos ^2 2 \alpha\)</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">\(\frac{2 u^2}{g} \cos ^2 \alpha\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\frac{2 u^2}{g} \cos ^2 \alpha\)</span> </div>

answer:
<div class="solution">According to the figure, the length of latus rectum is<br/>\(2(S M)=2 \times \frac{u^2}{2 g}(1+\cos 2 \alpha)=\frac{2 u^2 \cos ^2 \alpha}{g} .\)<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/KsiJvQnmYP5iFazmQ_rT-r9ryUB4O8eOkjKesxo9Odg.original.fullsize.png"/><br/></div>


tags: ['Mathematics', 'Parabola', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P2.db
title:
the-length-of-the-latusrectum-of-the-parabola-whose-focus-is-leftfracu22-g-sin-2-alphafracu22-g-cos-2-alpharight-and-directrix-is-yfracu22-g-is

slug:
the-length-of-the-latusrectum-of-the-parabola-whose-focus-is-leftfracu22-g-sin-2-alphafracu22-g-cos-2-alpharight-and-directrix-is-yfracu22-g-is-84006

question:
<div class="question">The length of the latus-rectum of the parabola whose focus is \(\left(\frac{u^2}{2 g} \sin 2 \alpha,-\frac{u^2}{2 g} \cos 2 \alpha\right)\) and directrix is \(y=\frac{u^2}{2 g}\), is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\frac{u^2}{g} \cos ^2 \alpha\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\frac{u^2}{g} \cos 2 \alpha\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(\frac{2 u^2}{g} \cos ^2 2 \alpha\)</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">\(\frac{2 u^2}{g} \cos ^2 \alpha\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\frac{2 u^2}{g} \cos ^2 \alpha\)</span> </div>

answer:
<div class="solution">According to the figure, the length of latus rectum is<br/>\(2(S M)=2 \times \frac{u^2}{2 g}(1+\cos 2 \alpha)=\frac{2 u^2 \cos ^2 \alpha}{g} .\)<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/KsiJvQnmYP5iFazmQ_rT-r9ryUB4O8eOkjKesxo9Odg.original.fullsize.png"/><br/></div>


title: the-length-of-the-perpendicular-drawn-from-the-point-3-1-11-to-the-line-2-x-3-y-2-4-z-3-is
slug: the-length-of-the-perpendicular-drawn-from-the-point-3-1-11-to-the-line-2-x-3-y-2-4-z-3-is-40075
tags: ['Mathematics', 'Three Dimensional Geometry', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P2.db
question:
<div class="question">The length of the perpendicular drawn from the point $(3,-1,11)$ to the line $\frac{x}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ is :</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\sqrt{29}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\sqrt{33}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\sqrt{53}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\sqrt{66}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\sqrt{53}$</span> </div>

answer:
<div class="solution">Let feet of perpendicular is
<br/> <br/>$(2 \alpha, 3 \alpha+2,4 \alpha+3)$
<br/> <br/>$\Rightarrow$ Direction ratio of the $\perp$ line is
<br/> <br/>$2 \alpha-3,3 \alpha+3,4 \alpha-8$. and
<br/> <br/>$\Rightarrow$ Direction ratio of the line are $2,3,4$
<br/> <br/>$\Rightarrow 2(2 \alpha-3)+3(3 \alpha+3)+4(4 \alpha-8)=0$
<br/> <br/>$\Rightarrow \alpha=1$
<br/> <br/>$\Rightarrow$ Feet of $\perp$ is $(2,5,7)$
<br/> <br/>$\Rightarrow$ Length $\perp$ is $\sqrt{1^{2}+6^{2}+4^{2}}=\sqrt{53}$</div>


tags: ['Mathematics', 'Circle', 'JEE Main']
source_db: MarksBatch2_P2.db
title:
the-length-of-the-tangent-drawn-from-any-point-on-the-circle-x-2-y-2-2-f-y-0-to-the-circle-x-2-y-2-2-fy-0-where-0-is-1

slug:
the-length-of-the-tangent-drawn-from-any-point-on-the-circle-x-2-y-2-2-f-y-0-to-the-circle-x-2-y-2-2-fy-0-where-0-is-1-23922

question:
<div class="question">The length of the tangent drawn from any point on the circle $x^{2}+y^{2}+2 f y+\lambda=0$ to the circle $\mathrm{x}^{2}+\mathrm{y}^{2}+2 \mathrm{fy}+\mu=0,$ where $\mu&gt;\lambda&gt;0,$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\sqrt{\mu-\lambda}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\sqrt{\mu+\lambda}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\sqrt{\mu^{2}-\lambda^{2}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mu+\lambda$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\sqrt{\mu-\lambda}$</span> </div>

answer:
<div class="solution">Let the radius of the first circle be $\mathrm{CT}=\mathrm{r}_{1}$. Also, let the radius of the second circle be $\mathrm{CP}=\mathrm{r}_{2}$
<br/>In the triangle $\mathrm{PCT}, \mathrm{T}$ is a right angle
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/Vx6mkKDSQTkMXkiFF9Z4ygG5UGK_UM5zCD04GOUxX60.original.fullsize.png"/><br/> <br/>$\begin{aligned} \text { So, } P T &amp;=\sqrt{P C^{2}-C T^{2}}=\sqrt{r_{1}^{2}-r_{2}^{2}} \\ &amp;=\sqrt{\left(f^{2}-\lambda\right)-\left(f^{2}-\mu\right)}=\sqrt{\mu-\lambda} \end{aligned}$</div>


tags: ['Mathematics', 'Circle', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P2.db
title:
the-length-of-the-tangent-drawn-from-any-point-on-the-circle-x-2-y-2-2-f-y-0-to-the-circle-x-2-y-2-2-fy-0-where-0-is

slug:
the-length-of-the-tangent-drawn-from-any-point-on-the-circle-x-2-y-2-2-f-y-0-to-the-circle-x-2-y-2-2-fy-0-where-0-is-60582

question:
<div class="question">The length of the tangent drawn from any point on the circle $x^{2}+y^{2}+2 f y+\lambda=0$ to the circle $\mathrm{x}^{2}+\mathrm{y}^{2}+2 \mathrm{fy}+\mu=0,$ where $\mu&gt;\lambda&gt;0,$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\sqrt{\mu-\lambda}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\sqrt{\mu+\lambda}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\sqrt{\mu^{2}-\lambda^{2}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mu+\lambda$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\sqrt{\mu-\lambda}$</span> </div>

answer:
<div class="solution">Let the radius of the first circle be $\mathrm{CT}=\mathrm{r}_{1}$. Also, let the radius of the second circle be $\mathrm{CP}=\mathrm{r}_{2}$
<br/>In the triangle $\mathrm{PCT}, \mathrm{T}$ is a right angle
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/Vx6mkKDSQTkMXkiFF9Z4ygG5UGK_UM5zCD04GOUxX60.original.fullsize.png"/><br/> <br/>$\begin{aligned} \text { So, } P T &amp;=\sqrt{P C^{2}-C T^{2}}=\sqrt{r_{1}^{2}-r_{2}^{2}} \\ &amp;=\sqrt{\left(f^{2}-\lambda\right)-\left(f^{2}-\mu\right)}=\sqrt{\mu-\lambda} \end{aligned}$</div>


tags: ['Mathematics', 'Circle', 'BITSAT', 'BITSAT 2016']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1: 2</span> </div>
source_db: MarksBatch2_P2.db
title:
the-lengths-of-the-tangent-drawn-from-any-point-on-the-circle-15-x-2-15-y-2-48-x-64-y-0-to-the-two-circles-5-x-2-5-y-2-24-x-32-y-75-0-and-5-x-2-5-y-2-

slug:
the-lengths-of-the-tangent-drawn-from-any-point-on-the-circle-15-x-2-15-y-2-48-x-64-y-0-to-the-two-circles-5-x-2-5-y-2-24-x-32-y-75-0-and-5-x-2-5-y-2-80996

question:
<div class="question">The lengths of the tangent drawn from any point
<br/> <br/>on the circle $15 x^{2}+15 y^{2}-48 x+64 y=0$ to the two circles $5 x^{2}+5 y^{2}-24 x+32 y+75=0$ and $5 x^{2}+5 y^{2}-48 x$ $+64 y+300=0$ are in the ratio of</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">1: 2</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2: 3</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">3: 4</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None</span> </li> </ul>

answer:
<div class="solution">Let \(\mathrm{P}(\mathrm{h}, \mathrm{k})\) be a point on the circle
<br/> <br/>\(15 x^{2}+15 y^{2}-48 x+64 y=0\)
<br/> <br/>Then the lengths of the tangents from \(\mathrm{P}(\mathrm{h}, \mathrm{k})\) to
<br/> <br/>\(\begin{aligned} &amp;5 x^{2}+5 y^{2}-24 x+32 y+75=0 \\ &amp;5 x^{2}+5 y^{2}-48 x+64 y+300=0 \text { are } \\ &amp;P T_{1}=\sqrt{h^{2}+k^{2}-\frac{24}{5} h+\frac{32}{5} k+15} \\ &amp;\text { and } P T_{2}=\sqrt{h^{2}+k^{2}-\frac{48}{5} h+\frac{64}{5} k+60} \\ &amp;\text { or } P T_{1}=\sqrt{\frac{48}{15} h-\frac{64}{15} k-\frac{24}{5} h+\frac{32}{5} k+15}=\sqrt{\frac{32}{15} k-\frac{24}{15} h+15} \end{aligned}\)
<br/> <br/>(Since \((\mathrm{h}, \mathrm{k})\) lies on \(15 \mathrm{x}^{2}-15 \mathrm{y}^{2}-48 \mathrm{x}+64 \mathrm{y}=0\)
<br/> <br/>\(\begin{aligned} &amp;\left.\therefore \mathrm{h}^{2}+\mathrm{k}^{2}-\frac{48}{15} \mathrm{~h}+\frac{64}{15} \mathrm{k}=0\right) \\ &amp;\text { and } \mathrm{PT}_{2}=\sqrt{\frac{48}{15} \mathrm{~h}-\frac{64}{5} \mathrm{k}-\frac{48}{5} \mathrm{~h}+\frac{64}{5} \mathrm{k}+60} \\ &amp;=\sqrt{-\frac{96}{15} \mathrm{~h}+\frac{128}{15} \mathrm{k}+60}=2 \sqrt{-\frac{24}{15} \mathrm{~h}+\frac{32}{15} \mathrm{k}+15}=2 \mathrm{PT}_{1} \\ &amp;\Rightarrow \mathrm{PT}_{1}: \mathrm{PT}_{2}=1: 2 \end{aligned}\)</div>


tags: ['Mathematics', 'Binomial Theorem', 'BITSAT', 'BITSAT 2009']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">89</span> </div>
source_db: MarksBatch2_P2.db
title:
the-letters-of-the-word-tough-are-written-in-all-possible-orders-and-these-words-are-written-out-as-in-a-dictionary-then-the-rank-of-the-word-tough-is

slug:
the-letters-of-the-word-tough-are-written-in-all-possible-orders-and-these-words-are-written-out-as-in-a-dictionary-then-the-rank-of-the-word-tough-is-97262

question:
<div class="question">The letters of the word TOUGH are written in all possible orders and these words are written out as in a dictionary, then the rank of the word TOUGH is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">120</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">88</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">89</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">90</span> </li> </ul>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; \text { Rank }=(4 ! \times 3)+(3 ! \times 2)+(2 ! \times 2)+1 \\<br/>&amp; =72+12+4+1=89<br/>\end{aligned}\)</div>


tags: ['Mathematics', 'Parabola', 'BITSAT', 'BITSAT 2009']
source_db: MarksBatch2_P2.db
title:
the-line-3-x4-y70-is-rotated-through-an-angle-fracpi4-in-the-clockwise-direction-about-the-point-11-the-equation-of-the-line-in-its-new-position-is

slug:
the-line-3-x4-y70-is-rotated-through-an-angle-fracpi4-in-the-clockwise-direction-about-the-point-11-the-equation-of-the-line-in-its-new-position-is-12426

question:
<div class="question">The line \(3 x-4 y+7=0\) is rotated through an angle \(\frac{\pi}{4}\) in the clockwise direction about the point \((-1,1)\). The equation of the line in its new position is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\(7 y+x-6=0\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(7 y-x-6=0\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(7 y+x+6=0\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(7 y-x+6=0\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(7 y+x-6=0\)</span> </div>

answer:
<div class="solution">As \((-1,1)\) is a point on \(3 x-4 y+7=0\), the rotation is possible. Slope of the given line \(=3 / 4\). Slope of the line in its new position<br/>\(=\frac{\frac{3}{4}-1}{1+\frac{3}{4}}=-\frac{1}{7}\)<br/>The required equation is<br/>\(\mathrm{y}-1=-\frac{1}{7}(\mathrm{x}+1) \text { or } 7 \mathrm{y}+\mathrm{x}-6=0\)</div>


title: the-line-a-xb-y1-cuts-ellipse-c-x2d-y21-only-once-if
slug: the-line-a-xb-y1-cuts-ellipse-c-x2d-y21-only-once-if-49051
question: <div class="question">The line \(a x+b y=1\) cuts ellipse \(c x^2+d y^2=1\) only once if</div>
tags: ['Mathematics', 'Circle', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P2.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\mathrm{ca}^2+\mathrm{db}^2=1\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\frac{\mathrm{c}}{\mathrm{a}^2}+\frac{\mathrm{d}}{\mathrm{b}^2}=1\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(\frac{\mathrm{a}^2}{\mathrm{c}}+\frac{\mathrm{b}^2}{\mathrm{~d}}=1\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(a c^2+b d^2=1\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\frac{\mathrm{a}^2}{\mathrm{c}}+\frac{\mathrm{b}^2}{\mathrm{~d}}=1\)</span> </div>

answer:
<div class="solution">Clearly \(a x+b y=1\)<br/>i.e \(y=-\frac{a}{b} x+\frac{1}{b}\) is tangent to<br/>\(\begin{aligned}<br/>&amp; \mathrm{cx}^2+\mathrm{dy}^2=1 \Rightarrow \frac{\mathrm{x}^2}{\frac{1}{c}}+\frac{\mathrm{y}^2}{\frac{1}{d}}=1 \\<br/>&amp; \therefore\left(\frac{1}{b}\right)^2=\left(\frac{1}{c}\right)\left(-\frac{a}{b}\right)^2+\left(\frac{1}{d}\right) \\<br/>&amp; \Rightarrow 1=\frac{a^2}{c}+\frac{b^2}{d}<br/>\end{aligned}\)</div>


tags: ['Mathematics', 'Circle', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch2_P2.db
title:
the-line-joining-50-to-10-cos-10-sin-is-divided-internally-in-the-ratio-2-3-at-p-if-varies-then-the-locus-of-p-is

slug:
the-line-joining-50-to-10-cos-10-sin-is-divided-internally-in-the-ratio-2-3-at-p-if-varies-then-the-locus-of-p-is-41856

question:
<div class="question">The line joining (5,0) to $((10 \cos \theta, 10 \sin \theta)$ is divided internally in the ratio 2: 3 at $P$. If $\theta$ varies, then the locus of $\mathrm{P}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">a pair of straight lines</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">a circle</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">a straight line</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">a circle</span> </div>

answer:
<div class="solution">Let $\mathrm{P}(\mathrm{x}, \mathrm{y})$ be the point dividing the join of $\mathrm{A}$ and $\mathrm{B}$ in the ratio 2: 3 internally, then
<br/> <br/>$$
<br/> <br/>\begin{array}{c}
<br/> <br/>\mathrm{x}=\frac{20 \cos \theta+15}{5}=4 \cos \theta+3 \Rightarrow \cos \theta=\frac{\mathrm{x}-3}{4} \\
<br/> <br/>\mathrm{y}=\frac{20 \sin \theta+0}{5}=4 \sin \theta \Rightarrow \sin \theta=\frac{\mathrm{y}}{4} \ldots \text { (ii) }
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>Squaring and adding (i) and (ii), we get the
<br/> <br/>required locus $(x-3)^{2}+y^{2}=16,$ which is
<br/> <br/>a circle.</div>


tags: ['Mathematics', 'Ellipse', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P3.db
title:
the-line-passing-through-the-extremity-a-of-the-major-axis-and-extremity-b-of-the-minor-axis-of-the-ellipse-x-2-9-y-2-9-meets-its-auxiliary-circle-at--1

slug:
the-line-passing-through-the-extremity-a-of-the-major-axis-and-extremity-b-of-the-minor-axis-of-the-ellipse-x-2-9-y-2-9-meets-its-auxiliary-circle-at-1-53803

question:
<div class="question">The line passing through the extremity <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> of the major axis and extremity <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> of the minor axis of the ellipse <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>9</mn><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>9</mn></math> meets its auxiliary circle at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math>. Then, the area of the triangle with vertices at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>,</mo><mo> </mo><mi>M</mi></math> and the origin <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>31</mn><mn>10</mn></mfrac></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>29</mn><mn>10</mn></mfrac></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>21</mn><mn>10</mn></mfrac></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>27</mn><mn>10</mn></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>27</mn><mn>10</mn></mfrac></math></span> </div>

answer:
<div class="solution"><p>Equation of auxiliary circle is</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>9</mn><mo> </mo><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mfenced><mi mathvariant="normal">i</mi></mfenced></math></p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/f750c76f-c25f-4349-91fd-82f7205a7d3f-d1.png" style="width: 288px; height: 230px;"/></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> Equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>M</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>x</mi><mn>3</mn></mfrac><mo>+</mo><mfrac><mi>y</mi><mn>1</mn></mfrac><mo>=</mo><mn>1</mn><mo> </mo><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mi>ii</mi></mfenced></math></p><p>On solving Eqs. (i) and (ii), we get</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mfenced><mrow><mo>-</mo><mfrac><mn>12</mn><mn>5</mn></mfrac><mo>,</mo><mfrac><mn>9</mn><mn>5</mn></mfrac></mrow></mfenced></math></p><p>Now, area of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Δ</mi><mi>A</mi><mi>O</mi><mi>M</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>O</mi><mi>A</mi><mo>×</mo><mi>M</mi><mi>N</mi><mo>=</mo><mfrac><mn>27</mn><mn>10</mn></mfrac></math> sq units</p><p>Hence, option (d) is correct.</p></div>


tags: ['Mathematics', 'Ellipse', 'JEE Main']
source_db: MarksBatch2_P3.db
title:
the-line-passing-through-the-extremity-a-of-the-major-axis-and-extremity-b-of-the-minor-axis-of-the-ellipse-x-2-9-y-2-9-meets-its-auxiliary-circle-at--2

slug:
the-line-passing-through-the-extremity-a-of-the-major-axis-and-extremity-b-of-the-minor-axis-of-the-ellipse-x-2-9-y-2-9-meets-its-auxiliary-circle-at-2-19485

question:
<div class="question">The line passing through the extremity <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> of the major axis and extremity <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> of the minor axis of the ellipse <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>9</mn><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>9</mn></math> meets its auxiliary circle at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math>. Then, the area of the triangle with vertices at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>,</mo><mo> </mo><mi>M</mi></math> and the origin <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>31</mn><mn>10</mn></mfrac></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>29</mn><mn>10</mn></mfrac></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>21</mn><mn>10</mn></mfrac></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>27</mn><mn>10</mn></mfrac></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>27</mn><mn>10</mn></mfrac></math></span> </div>

answer:
<div class="solution"><p>Equation of auxiliary circle is</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>9</mn><mo> </mo><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mfenced><mi mathvariant="normal">i</mi></mfenced></math></p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/f750c76f-c25f-4349-91fd-82f7205a7d3f-d1.png" style="width: 288px; height: 230px;"/></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> Equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>M</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>x</mi><mn>3</mn></mfrac><mo>+</mo><mfrac><mi>y</mi><mn>1</mn></mfrac><mo>=</mo><mn>1</mn><mo> </mo><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mi>ii</mi></mfenced></math></p><p>On solving Eqs. (i) and (ii), we get</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mfenced><mrow><mo>-</mo><mfrac><mn>12</mn><mn>5</mn></mfrac><mo>,</mo><mfrac><mn>9</mn><mn>5</mn></mfrac></mrow></mfenced></math></p><p>Now, area of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Δ</mi><mi>A</mi><mi>O</mi><mi>M</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>O</mi><mi>A</mi><mo>×</mo><mi>M</mi><mi>N</mi><mo>=</mo><mfrac><mn>27</mn><mn>10</mn></mfrac></math> sq units</p><p>Hence, option (d) is correct.</p></div>


title: the-line-which-is-parallel-to-xaxis-and-crosses-the-curve-y-x-at-an-angle-of-4-5-is
slug: the-line-which-is-parallel-to-xaxis-and-crosses-the-curve-y-x-at-an-angle-of-4-5-is-92656
tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P3.db
question:
<div class="question">The line which is parallel to X-axis and crosses the curve $y=\sqrt{x}$ at an angle of $45^{\circ}$, is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x=\frac{1}{4}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$y=\frac{1}{4}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$y=\frac{1}{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$y=1$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$y=\frac{1}{2}$</span> </div>

answer:
<div class="solution">Given equation of a line parallel to X-axis is $\mathrm{y}=\mathrm{k}$
<br/> <br/>Given equation of the curve is $y=\sqrt{x}$, On solving equation of line with the equation of curve, we get $x=k^{2}$ Thus the intersecting point is $\left(\mathrm{k}^{2}, \mathrm{k}\right)$ It is given that the line $\mathrm{y}=\mathrm{k}$ intersect the curve $y=\sqrt{x}$ at an angle of $\pi / 4$. This means that the slope of the tangent to
<br/> <br/>$$
<br/> <br/>\mathrm{y}=\sqrt{\mathrm{x}} \text { at }\left(\mathrm{k}^{2}, \mathrm{k}\right) \text { is } \tan \left(\pm \frac{\pi}{4}\right)=\pm 1
<br/> <br/>$$
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>\Rightarrow\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)_{\left(\mathrm{k}^{2}, \mathrm{k}\right)}=\pm 1 \Rightarrow\left(\frac{1}{2 \sqrt{\mathrm{x}}}\right)_{\left(\mathrm{k}^{2}, \mathrm{k}\right)}=\pm 1 \\
<br/> <br/>\Rightarrow \mathrm{k}=\pm \frac{1}{2}
<br/> <br/>\end{array}
<br/> <br/>$$</div>


tags: ['Mathematics', 'Area Under Curves', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch2_P3.db
title:
the-line-y-m-x-bisects-the-area-enclosed-by-lines-x-0-y-0-and-x-32-and-the-curve-y-1-4-x-x-2-then-the-value-of-m-is

slug:
the-line-y-m-x-bisects-the-area-enclosed-by-lines-x-0-y-0-and-x-32-and-the-curve-y-1-4-x-x-2-then-the-value-of-m-is-78441

question:
<div class="question">The line $y=m x$ bisects the area enclosed by lines $x$ $=0, \mathrm{y}=0$ and $\mathrm{x}=3 / 2$ and the curve $\mathrm{y}=1+4 \mathrm{x}-\mathrm{x}^{2}$
<br/> <br/>Then the value of $\mathrm{m}$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{13}{6}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{13}{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{13}{5}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{13}{7}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{13}{6}$</span> </div>

answer:
<div class="solution">$y=1+4 x-x^{2}=5-(x-2)^{2}$
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/MzY8V4T1MJrhjGUbkQlVjCwugtbohShak5aX8iphZQQ.original.fullsize.png"/><br/> <br/> <br/>We have $\int_{0}^{3 / 2}\left(1+4 x-x^{2}\right) \mathrm{dx}=2 \int_{0}^{3 / 2} \operatorname{mx} \mathrm{dx}$
<br/> <br/>$=\frac{3}{2}+2\left(\frac{9}{4}\right)-\frac{1}{3}\left(\frac{27}{8}\right)=\mathrm{m} \cdot \frac{9}{4}$
<br/> <br/>On solving we get $\mathrm{m}=\frac{13}{6}$</div>


title: the-lines-1-x-2-1-y-3-k-z-4-and-k-x-1-2-y-4-1-z-5-are-coplanar-if
slug: the-lines-1-x-2-1-y-3-k-z-4-and-k-x-1-2-y-4-1-z-5-are-coplanar-if-83033
tags: ['Mathematics', 'Three Dimensional Geometry', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">The lines $\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{-k}$ and $\frac{x-1}{k}=\frac{y-4}{2}=\frac{z-5}{1}$ are coplanar if</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{k}=3$ or -2</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{k}=0$ or -1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{k}=1$ or -1</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\mathrm{k}=0$ or -3</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{k}=0$ or -3</span> </div>

answer:
<div class="solution">Two planes are coplanar if<br/>$\left|\begin{array}{ccc}x_2-x_1 &amp; y_2-y_1 &amp; z_2-z_1 \\ l_1 &amp; m_1 &amp; n_1 \\ l_2 &amp; m_2 &amp; n_2\end{array}\right|=0$<br/>$\left|\begin{array}{ccc}1 &amp; -1 &amp; -1 \\ 1 &amp; 1 &amp; -k \\ k &amp; 2 &amp; 1\end{array}\right|=0$<br/>Applying $\mathrm{C}_2 \rightarrow \mathrm{C}_2+\mathrm{C}_1, \mathrm{C}_3 \rightarrow \mathrm{C}_3+\mathrm{C}_1$<br/>$\Rightarrow\left|\begin{array}{ccc}1 &amp; 0 &amp; 0 \\ 1 &amp; 2 &amp; 1-k \\ k &amp; k+2 &amp; 1+k\end{array}\right|=0$<br/>$\Rightarrow \quad 1[2+2 k-(k+2)(1-k)]=0$<br/>$\Rightarrow \quad 2+2 k-\left(-k^2-k+2\right)=0$<br/>$k^2+3 k=0 \Rightarrow k(k+3)=0$</div>


title: the-load-versus-elongation-graph-for-four-wires-is-shown-the-thinnest-wire-is-1
slug: the-load-versus-elongation-graph-for-four-wires-is-shown-the-thinnest-wire-is-1-79292
tags: ['Physics', 'Mechanical Properties of Solids', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$S$</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">The load versus elongation graph for four wires is shown. The thinnest wire is
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/FNqgPKOYJI4uU0XLo-RiCkRNhHbbnFof48miPIa87R0.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$P$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$Q$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$R$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$S$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Using \(\quad \mathrm{Y}=\frac{\mathrm{F} / \mathrm{A}}{\Delta \mathrm{L} / \mathrm{L}}\)
<br/> <br/>Cross-section area of the wire \(\quad \mathrm{A}=\pi \mathrm{R}^{2}\) where \(\quad \mathrm{R}\) is the radius of the wire \(\therefore \quad \mathrm{Y}=\frac{\mathrm{F}}{\pi \mathrm{R}^{2}} \times \frac{\mathrm{L}}{\Delta \mathrm{L}}\)
<br/> <br/>\(\Longrightarrow \mathrm{F}=\frac{\pi \mathrm{R}^{2} \mathrm{Y}}{\mathrm{L}} \Delta \mathrm{L}\)
<br/> <br/>Thus Slope of load vs elongation graph, \(\mathrm{s}=\frac{\pi \mathrm{R}^{2} \mathrm{Y}}{\mathrm{L}}\)
<br/> <br/>\(\Longrightarrow \quad \mathrm{s} \propto \mathrm{R}^{2}\)
<br/> <br/>Now from the graph shown, OA has the least slope and hence the least radius. Thus \(\mathrm{OA}\) represents the thinnest wire.</div>


tags: ['Mathematics', 'Ellipse', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P3.db
title:
the-locus-of-the-foot-of-perpendicular-drawn-from-the-centre-of-the-ellipse-x-2-3-y-2-6-on-any-tangent-to-it-is-1

slug:
the-locus-of-the-foot-of-perpendicular-drawn-from-the-centre-of-the-ellipse-x-2-3-y-2-6-on-any-tangent-to-it-is-1-54817

question:
<div class="question">The locus of the foot of perpendicular drawn from the centre of the ellipse <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>6</mn></math> on any tangent to it is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>y</mi><mn>2</mn></msup></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><msup><mi>y</mi><mn>2</mn></msup></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>y</mi><mn>2</mn></msup></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><msup><mi>y</mi><mn>2</mn></msup></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><msup><mi>y</mi><mn>2</mn></msup></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><msup><mi>y</mi><mn>2</mn></msup></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><msup><mi>y</mi><mn>2</mn></msup></math></span> </div>

answer:
<div class="solution"><blockquote><p>We have,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>6</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>6</mn></mfrac><mo>+</mo><mfrac><msup><mi>y</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>=</mo><mn>1</mn></math></p><p>Centre of an ellipse is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>,</mo><mn>0</mn></mrow></mfenced></math>.</p><p>Let the foot of perpendicular be <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>h</mi><mo>,</mo><mi>k</mi><mo>)</mo></math>.</p><p>Equation of tangent in slope form to the standard ellipse <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mi>x</mi><mn>2</mn></msup><msup><mi>a</mi><mn>2</mn></msup></mfrac><mo>+</mo><mfrac><msup><mi>y</mi><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup></mfrac><mo>=</mo><mn>1</mn></math> is</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>m</mi><mi>x</mi><mo>±</mo><msqrt><msup><mi>a</mi><mn>2</mn></msup><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></msqrt></math>.</p><p>Hence, equation of tangent with slope <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> passing <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>h</mi><mo>,</mo><mi>k</mi><mo>)</mo></math> is </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>m</mi><mi>x</mi><mo>±</mo><msqrt><mn>6</mn><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></msqrt></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>k</mi><mo>=</mo><mi>m</mi><mi>h</mi><mo>±</mo><msqrt><mn>6</mn><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></msqrt></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mo>-</mo><mfrac><mi>h</mi><mi>k</mi></mfrac></math>.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>k</mi><mo>=</mo><mi>h</mi><mfenced><mrow><mo>-</mo><mfrac><mi>h</mi><mi>k</mi></mfrac></mrow></mfenced><mo>±</mo><msqrt><mn>6</mn><msup><mi>m</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></msqrt></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msqrt><mfrac><mrow><mn>6</mn><msup><mi>h</mi><mn>2</mn></msup></mrow><msup><mi>k</mi><mn>2</mn></msup></mfrac><mo>+</mo><mn>2</mn></msqrt><mo>=</mo><mfrac><mrow><msup><mi>h</mi><mn>2</mn></msup><mo>+</mo><msup><mi>k</mi><mn>2</mn></msup></mrow><mi>k</mi></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>6</mn><msup><mi>h</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><msup><mi>k</mi><mn>2</mn></msup><mo>=</mo><msup><mfenced><mrow><msup><mi>h</mi><mn>2</mn></msup><mo>+</mo><msup><mi>k</mi><mn>2</mn></msup></mrow></mfenced><mn>2</mn></msup></math></p><p>So, required locus is</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mrow></mfenced><mn>2</mn></msup></math>.</p></blockquote></div>


tags: ['Mathematics', 'Circle', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P3.db
title:
the-locus-of-the-midpoint-of-the-chord-if-contact-of-tangents-drawn-from-points-lying-on-the-straight-line-4-x-5-y-20-to-the-circle-x-2-y-2-9-is

slug:
the-locus-of-the-midpoint-of-the-chord-if-contact-of-tangents-drawn-from-points-lying-on-the-straight-line-4-x-5-y-20-to-the-circle-x-2-y-2-9-is-24460

question:
<div class="question">The locus of the mid-point of the chord if contact of tangents drawn from points lying on the straight line $4 x-5 y=20$ to the circle $x^2+y^2=9$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"> <br/>$20\left(x^2+y^2\right)-36 x+45 y=0$
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <br/>$20\left(x^2+y^2\right)+36 x-45 y=0$
<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>$36\left(x^2+y^2\right)-20 x+45 y=0$
<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>$36\left(x^2+y^2\right)+20 x-45 y=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$20\left(x^2+y^2\right)-36 x+45 y=0$
<br/></span> </div>

answer:
<div class="solution">Let $P\left(t, \frac{4 t-20}{5}\right)$ be a point on the line
<br/>$4 x-5 y=20$. Then the chord of contact of tangents drawn from $P$ to the circle $x^2+y^2=9$ is
<br/>$t x+\left(\frac{4 t-20}{5}\right) y=9\quad \quad \dots(i)$
<br/>Let $\theta(h, k)$ be the mid-point of this chord of contact, then its equation is also $h x+k y=h^2+k^2\quad \quad \dots(ii)$
<br/>$\left[\right.$ using $\left.T=S^{\prime}\right]$
<br/>Clearly Eqs. (i) and (ii) represent the same line
<br/>$\begin{aligned}
<br/>&amp; \therefore \frac{t}{h}=\frac{4 t-20}{5 k}=\frac{9}{h^2+k^2} \\
<br/>&amp; \Rightarrow \frac{t}{h}=\frac{9}{h^2+k^2} \text { and } \frac{t}{h}=\frac{4 t-20}{5 k} \\
<br/>&amp; \Rightarrow t=\frac{9 h}{h^2+k^2} \text { and } t=\frac{20 h}{4 h-5 k} \\
<br/>&amp; \Rightarrow \frac{9 h}{h^2+k^2}=\frac{20 h}{4 h-5 k} \\
<br/>&amp; \Rightarrow  h\left\{20\left(h^2+k^2\right)-36 h+45 k\right\}=0 \\
<br/>&amp; x=0, \text { or }\left[20\left(x^2+y^2\right)-36 x+45 y\right]=0
<br/>\end{aligned}$</div>


tags: ['Mathematics', 'Parabola', 'JEE Main']
source_db: MarksBatch2_P3.db
title:
the-locus-of-the-point-of-intersection-of-two-tangents-to-the-parabola-y-2-4-a-x-which-are-at-right-angle-to-one-another-is-1

slug:
the-locus-of-the-point-of-intersection-of-two-tangents-to-the-parabola-y-2-4-a-x-which-are-at-right-angle-to-one-another-is-1-52121

question:
<div class="question">The locus of the point of intersection of two tangents to the parabola $y^{2}=4 a x,$ which are at right angle to one another is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x^{2}+y^{2}=a^{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$a y^{2}=x$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$x+a=0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x+y \pm a=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x+a=0$</span> </div>

answer:
<div class="solution">Let the two tangents to the parabola \(y^{2}=4 a x\) be \(P T\) and \(Q T\) which are at right angle to one another at \(\mathrm{T}(\mathrm{h}, \mathrm{k})\).
<br/> <br/>Then we have a find the locus of \(\mathrm{T}(\mathrm{h}, \mathrm{k})\).
<br/> <br/>We know that \(y=m x+\frac{a}{m}\), where \(m\) is the slope is the equation of tangent to the parabola \(\mathrm{y}^{2}=4 \mathrm{ax}\) for all \(\mathrm{m}\).
<br/> <br/>Since this tangent to the parabola will pass through \(\mathrm{T}(\mathrm{h}, \mathrm{k})\), so
<br/> <br/>\(\mathrm{k}=\mathrm{mh}+\frac{\mathrm{a}}{\mathrm{m}} ; \text { or } \mathrm{m}^{2} \mathrm{~h}-\mathrm{mk}+\mathrm{a}=0\)
<br/> <br/>This is a quadratic equation in \(m\), so will have two roots, say \(m_{1}\) and \(m_{2}\), then \(\mathrm{m}_{1}+\mathrm{m}_{2}=\frac{\mathrm{k}}{\mathrm{h}}\), and \(\mathrm{m}_{1} \cdot \mathrm{m}_{2}=\frac{\mathrm{a}}{\mathrm{h}}\)
<br/> <br/>Given that the two tangents intersect at right angle so
<br/> <br/>\(\mathrm{m}_{1} \cdot \mathrm{m}_{2}=-1 \text { or } \frac{\mathrm{a}}{\mathrm{h}}=-1 \text { or } \mathrm{h}+\mathrm{a}=0\)
<br/> <br/>The locus of \(\mathrm{T}(\mathrm{h}, \mathrm{k})\) is \(\mathrm{x}+\mathrm{a}=0\), which is the equation of directrix.
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/ehwyyskwi38789.png"/></div>


tags: ['Mathematics', 'Parabola', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch2_P3.db
title:
the-locus-of-the-point-of-intersection-of-two-tangents-to-the-parabola-y-2-4-a-x-which-are-at-right-angle-to-one-another-is

slug:
the-locus-of-the-point-of-intersection-of-two-tangents-to-the-parabola-y-2-4-a-x-which-are-at-right-angle-to-one-another-is-42249

question:
<div class="question">The locus of the point of intersection of two tangents to the parabola $y^{2}=4 a x,$ which are at right angle to one another is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x^{2}+y^{2}=a^{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$a y^{2}=x$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$x+a=0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x+y \pm a=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x+a=0$</span> </div>

answer:
<div class="solution">Let the two tangents to the parabola \(y^{2}=4 a x\) be \(P T\) and \(Q T\) which are at right angle to one another at \(\mathrm{T}(\mathrm{h}, \mathrm{k})\).
<br/> <br/>Then we have a find the locus of \(\mathrm{T}(\mathrm{h}, \mathrm{k})\).
<br/> <br/>We know that \(y=m x+\frac{a}{m}\), where \(m\) is the slope is the equation of tangent to the parabola \(\mathrm{y}^{2}=4 \mathrm{ax}\) for all \(\mathrm{m}\).
<br/> <br/>Since this tangent to the parabola will pass through \(\mathrm{T}(\mathrm{h}, \mathrm{k})\), so
<br/> <br/>\(\mathrm{k}=\mathrm{mh}+\frac{\mathrm{a}}{\mathrm{m}} ; \text { or } \mathrm{m}^{2} \mathrm{~h}-\mathrm{mk}+\mathrm{a}=0\)
<br/> <br/>This is a quadratic equation in \(m\), so will have two roots, say \(m_{1}\) and \(m_{2}\), then \(\mathrm{m}_{1}+\mathrm{m}_{2}=\frac{\mathrm{k}}{\mathrm{h}}\), and \(\mathrm{m}_{1} \cdot \mathrm{m}_{2}=\frac{\mathrm{a}}{\mathrm{h}}\)
<br/> <br/>Given that the two tangents intersect at right angle so
<br/> <br/>\(\mathrm{m}_{1} \cdot \mathrm{m}_{2}=-1 \text { or } \frac{\mathrm{a}}{\mathrm{h}}=-1 \text { or } \mathrm{h}+\mathrm{a}=0\)
<br/> <br/>The locus of \(\mathrm{T}(\mathrm{h}, \mathrm{k})\) is \(\mathrm{x}+\mathrm{a}=0\), which is the equation of directrix.
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/ehwyyskwi38789.png"/></div>


title: the-logic-operations-performed-by-the-given-digital-circuit-is-equivalent-to-1
slug: the-logic-operations-performed-by-the-given-digital-circuit-is-equivalent-to-1-47777
tags: ['Physics', 'Semiconductors', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">The logic operations performed by the given digital circuit is equivalent to:<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/BXaO_1eHuJsLMucprGNkUpNXeKEp53ZePPE4tLgOYQ4.original.fullsize.png"/><br/></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><br/>AND</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>NOR</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>OR</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>NAND</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>AND</span> </div>

answer:
<div class="solution"><br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/qVK9FXd8k-4NE2hDvgG2qByCP2qAhE1Tjun1aGX-xP4.original.fullsize.png"/><br/><br/>$\begin{aligned} &amp; \mathrm{Y}=\overline{\mathrm{Z}}=(\mathrm{A}+\mathrm{B}) \cdot(\mathrm{A} \cdot \mathrm{B})=\mathrm{A} \cdot \mathrm{A} \cdot \mathrm{B}+\mathrm{AB} \cdot \mathrm{B} \\ &amp; \mathrm{Y}=\mathrm{AB}+\mathrm{AB}=\mathrm{A} \cdot \mathrm{B} \\ &amp; \therefore \text { It is an } \mathrm{AND} \text { Gate. }\end{aligned}$</div>


title: the-magnetic-field-at-the-centre-o-of-the-currentcarrying-square-loop-shown-in-the-figure-is
slug: the-magnetic-field-at-the-centre-o-of-the-currentcarrying-square-loop-shown-in-the-figure-is-22808
tags: ['Physics', 'Magnetic Effects of Current', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P3.db
question:
<div class="question"><p>The magnetic field at the centre <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi></math> of the current-carrying square loop shown in the figure is</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/5365d98f-4dbd-4b85-b8c5-b71024247ff3-image.png" style="width: 140px; height: 183px;"/></p></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>5</mn></mrow></msup><mo> </mo><mi mathvariant="normal">T</mi></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>5</mn></mrow></msup><mo> </mo><mi mathvariant="normal">T</mi></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><msqrt><mn>2</mn></msqrt><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>5</mn></mrow></msup><mo> </mo><mi mathvariant="normal">T</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msqrt><mn>2</mn></msqrt><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>5</mn></mrow></msup><mo> </mo><mi mathvariant="normal">T</mi></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><msqrt><mn>2</mn></msqrt><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>5</mn></mrow></msup><mo> </mo><mi mathvariant="normal">T</mi></math></span> </div>

answer:
<div class="solution"><p>Given, current through the square loop is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mi mathvariant="normal">A</mi></math>.</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/fd93c099-a7a1-4a47-82b0-d8529d10f77c-image.png" style="width: 200px; height: 207px;"/></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mo>=</mo><mn>2</mn><mo> </mo><mi>A</mi></math></p><p>The given situation is shown in the figure. Magnetic field due to wire <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>B</mi></math> at the centre <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi></math> is</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>B</mi><mrow><mi>A</mi><mi>B</mi></mrow></msub><mo>=</mo><mfrac><msub><mi>μ</mi><mn>0</mn></msub><mrow><mn>4</mn><mi>π</mi></mrow></mfrac><mo>·</mo><mfrac><mi>I</mi><mi>r</mi></mfrac><mfenced close="]" open="["><mrow><mi>sin</mi><mn>45</mn><mo>°</mo><mo>+</mo><mi>sin</mi><mn>45</mn><mo>°</mo></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>7</mn></mrow></msup><mo>×</mo><mfrac><mn>2</mn><mrow><mn>2</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfrac><mfenced close="]" open="["><mrow><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac><mo>+</mo><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msqrt><mn>2</mn></msqrt><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>5</mn></mrow></msup><mo> </mo><mi mathvariant="normal">T</mi></math> [Downward to the plane of square loop]</p><p>Since, magnetic field due to side <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>B</mi><mo>,</mo><mo> </mo><mi>B</mi><mi>C</mi><mo>,</mo><mo> </mo><mi>C</mi><mi>D</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mi>A</mi></math> is same in magnitude and direction.</p><p>Hence, net magnetic field at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>=</mo><mn>4</mn><mo>×</mo><msub><mi>B</mi><mrow><mi>A</mi><mi>B</mi></mrow></msub><mo>=</mo><mn>4</mn><msqrt><mn>2</mn></msqrt><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>5</mn></mrow></msup><mo> </mo><mi mathvariant="normal">T</mi></math></p></div>


title: the-magnetic-field-at-the-origin-due-to-the-current-flowing-in-the-wire-is-1
slug: the-magnetic-field-at-the-origin-due-to-the-current-flowing-in-the-wire-is-1-32405
tags: ['Physics', 'Magnetic Effects of Current', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P3.db
question:
<div class="question">The magnetic field at the origin due to the current flowing in the wire is $-$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/rzU5BzCSnMH5evuZOfWyWmjTQsstco0Xd7nKEASEnwE.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-\frac{\mu_{0} \mathrm{I}}{8 \pi \mathrm{a}}(\hat{\mathrm{i}}+\hat{\mathrm{k}})$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\mu_{0} \mathrm{I}}{2 \pi \mathrm{a}}(\hat{\mathrm{i}}+\hat{\mathrm{k}})$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{\mu_{0} \mathrm{I}}{8 \pi \mathrm{a}}(-\hat{\mathrm{i}}+\hat{\mathrm{k}})$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\mu_{0} \mathrm{I}}{4 \pi \mathrm{a} \sqrt{2}}(\hat{\mathrm{i}}-\hat{\mathrm{k}})$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\mu_{0} \mathrm{I}}{8 \pi \mathrm{a}}(-\hat{\mathrm{i}}+\hat{\mathrm{k}})$</span> </div>

answer:
<div class="solution">$\mathrm{B}_{\mathrm{OD}}=0, \mathrm{~B}_{\mathrm{OB}}=0$<br/>$\mathrm{~B}_{\mathrm{AB}}=\frac{\mu_{0} \mathrm{I}}{4 \pi \mathrm{a} \sqrt{2}}\left[\cos 45^{\circ}(-\hat{\mathrm{i}})+\cos 45^{\circ} \hat{\mathrm{k}}\right]$<br/>$=\frac{\mu_{0} \mathrm{I}}{8 \pi \mathrm{a}}(-\hat{\mathrm{i}}+\hat{\mathrm{k}})$</div>


tags: ['Physics', 'Electromagnetic Waves', 'BITSAT', 'BITSAT 2009']
source_db: MarksBatch2_P3.db
title:
the-magnetic-field-in-a-travelling-electromagnetic-wave-has-a-peak-value-of-20-nt-the-peak-value-of-electric-field-strength-is-2

slug:
the-magnetic-field-in-a-travelling-electromagnetic-wave-has-a-peak-value-of-20-nt-the-peak-value-of-electric-field-strength-is-2-37679

question:
<div class="question">The magnetic field in a travelling electromagnetic wave has a peak value of $20 \mathrm{nT}$. The peak value of electric field strength is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$3 \mathrm{~V} / \mathrm{m}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$6 \mathrm{~V} / \mathrm{m}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$9 \mathrm{~V} / \mathrm{m}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$12 \mathrm{~V} / \mathrm{m}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$6 \mathrm{~V} / \mathrm{m}$</span> </div>

answer:
<div class="solution">$\overrightarrow{\mathrm{E}}_{0}=\overrightarrow{\mathrm{B}}_{0} \times \overrightarrow{\mathrm{C}}$<br/>$$<br/>\left|\overrightarrow{\mathrm{E}}_{0}\right|=\left|\mathrm{B}_{0}\right| \cdot|\overrightarrow{\mathrm{C}}|=20 \times 10^{-9} \times 3 \times 10^{8}=6 \mathrm{~V} / \mathrm{m}<br/>$$</div>


tags: ['Physics', 'Magnetic Effects of Current', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
source_db: MarksBatch2_P3.db
title:
the-magnetic-moment-of-an-electron-e-revolving-in-an-orbit-around-nucleus-with-an-orbital-angular-momentum-is-given-by-1

slug:
the-magnetic-moment-of-an-electron-e-revolving-in-an-orbit-around-nucleus-with-an-orbital-angular-momentum-is-given-by-1-36829

question:
<div class="question">The magnetic moment of an electron (e) revolving in an orbit around nucleus with an orbital angular momentum is given by:</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\vec{\mu}_L=\frac{e \vec{L}}{2 m}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\vec{\mu}_L=-\frac{e \vec{L}}{2 m}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\vec{\mu}_l=-\frac{e \vec{L}}{m}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\vec{\mu}_l=\frac{2 e \vec{L}}{m}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\vec{\mu}_L=\frac{e \vec{L}}{2 m}$</span> </div>

answer:
<div class="solution">As $\overrightarrow{\mathrm{M}}=\mathrm{IA}$<br/>$\Rightarrow|\overrightarrow{\mathrm{M}}|=\frac{\mathrm{e}}{\frac{2 \pi \mathrm{R}}{\mathrm{v}}} \pi \mathrm{R}^2$ $\left[\because \mathrm{I}=\frac{\mathrm{Q}}{\mathrm{T}}=\frac{\mathrm{e}}{\frac{2 \pi \mathrm{R}}{\mathrm{V}}}\right]$<br/>$\Rightarrow|\vec{M}|=\frac{1}{2} e v R \Rightarrow|\vec{M}|=\frac{\mathrm{mvR}}{1} \cdot \frac{\mathrm{e}}{2 \mathrm{~m}}$<br/>$\Rightarrow|\overrightarrow{\mathrm{M}}|=\frac{\mathrm{eL}}{2 \mathrm{~m}} \Rightarrow|\overrightarrow{\mathrm{M}}|=-\frac{\mathrm{e} \overrightarrow{\mathrm{L}}}{2 \mathrm{~m}}$<br/>[ $\because$ Here $\overrightarrow{\mathrm{M}}$ and $\overrightarrow{\mathrm{L}}$ will always be opposite]</div>


tags: ['Physics', 'Magnetic Effects of Current', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch2_P3.db
title:
the-magnetic-moments-associated-with-two-closely-wound-circular-coils-a-and-b-of-radius-r-a-10-cm-and-r-b-20-cm-respectively-are-equal-if-where-n-a-i--1

slug:
the-magnetic-moments-associated-with-two-closely-wound-circular-coils-a-and-b-of-radius-r-a-10-cm-and-r-b-20-cm-respectively-are-equal-if-where-n-a-i-1-52735

question:
<div class="question">The magnetic moments associated with two closely wound circular coils A and B of radius $r_A=10 \mathrm{~cm}$ and $r_B=20 \mathrm{~cm}$ respectively are equal if: (Where $\mathrm{N}_A, \mathrm{I}_A$ and $\mathrm{N}_{\mathrm{B}}, \mathrm{I}_{\mathrm{B}}$ are number of turn and current of $A$ and $B$ respectively)</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2 \mathrm{~N}_{\mathrm{A}} \mathrm{I}_{\mathrm{A}}=\mathrm{N}_{\mathrm{B}} \mathrm{I}_{\mathrm{B}}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{N}_{\mathrm{A}}=2 \mathrm{~N}_{\mathrm{B}}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\mathrm{N}_{\mathrm{A}} \mathrm{I}_{\mathrm{A}}=4 \mathrm{~N}_{\mathrm{B}} \mathrm{I}_{\mathrm{B}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$4 \mathrm{~N}_{\mathrm{A}} \mathrm{I}_{\mathrm{A}}=\mathrm{N}_{\mathrm{B}} \mathrm{I}_{\mathrm{B}}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{N}_{\mathrm{A}} \mathrm{I}_{\mathrm{A}}=4 \mathrm{~N}_{\mathrm{B}} \mathrm{I}_{\mathrm{B}}$</span> </div>

answer:
<div class="solution">The magnetic moment associated with circular coil,<br/>$\begin{aligned}<br/>&amp; M=N I A \\<br/>&amp; M_A=M_B \\<br/>&amp; \therefore N_A I_A A_A=N_B I_B A_B \\<br/>&amp; \therefore N_A I_A \pi(0.1)^2=N_B I_B \pi(0.2)^2\left(\because A=\pi r^2\right) \\<br/>&amp; \therefore N_A I_A=4 N_B I_B<br/>\end{aligned}$</div>


tags: ['Physics', 'Ray Optics', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P3.db
title:
the-magnifying-power-of-a-telescope-is-9-when-it-is-adjusted-for-parallel-rays-the-distance-between-the-objective-and-the-eye-piece-is-found-to-be-20-

slug:
the-magnifying-power-of-a-telescope-is-9-when-it-is-adjusted-for-parallel-rays-the-distance-between-the-objective-and-the-eye-piece-is-found-to-be-20-23278

question:
<div class="question">The magnifying power of a telescope is $9 .$ When it is adjusted for parallel rays, the distance between the objective and the eye piece is found to be $20 \mathrm{~cm}$. The focal length of lenses are</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$18 \mathrm{~cm}, 2 \mathrm{~cm}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$11 \mathrm{~cm}, 9 \mathrm{~cm}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$10 \mathrm{~cm}, 10 \mathrm{~cm}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$15 \mathrm{~cm}, 5 \mathrm{~cm}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$18 \mathrm{~cm}, 2 \mathrm{~cm}$</span> </div>

answer:
<div class="solution">$\frac{\mathrm{f}_{0}}{\mathrm{f}_{\mathrm{e}}}=9, \quad \therefore \mathrm{f}_{0}=9 \mathrm{f}_{\mathrm{e}}$
<br/> <br/>Also $\mathrm{f}_{0}+\mathrm{f}_{\mathrm{e}}=20(\because$ final image is at infinity $)$ $9 \mathrm{f}_{\mathrm{e}}+\mathrm{f}_{\mathrm{e}}=20, \mathrm{f}_{\mathrm{e}}=2 \mathrm{~cm}, \quad \therefore \mathrm{f}_{0}=18 \mathrm{~cm}$</div>


title: the-major-product-y-in-the-following-reaction-is-3
slug: the-major-product-y-in-the-following-reaction-is-3-79596
tags: ['Chemistry', 'Haloalkanes and Haloarenes', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P3.db
question:
<div class="question">The major product ' $\mathrm{Y}^{\prime}$ in the following reaction is :
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/C9oI33_312rIoiawGBuD0u-U_FIXN-rrtBY_oUaUgjk.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/MrqQzmtqhW5_F1D5ndZgoxf4h4Kp_ZArm_LKRKqLLxc.original.fullsize.png"/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/IpWhXIciGK6rQTYFFqs-l5thhHXbljgcYd-DN5AaLhY.original.fullsize.png"/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/sZOqBxaX63aSfgIPz87aglaf6_y4vLGOz3lN4wHcc6A.original.fullsize.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/82I1Gzjt9ea3Orftxk3zpu3oraKxE3B49gNMURc4Xe4.original.fullsize.png"/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/sZOqBxaX63aSfgIPz87aglaf6_y4vLGOz3lN4wHcc6A.original.fullsize.png"/></span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/y1ruUnRcXk0PwcMJCFabqasUqP14WhBzlwv04Zbvy1Y.original.fullsize.png"/></div>


tags: ['Physics', 'Nuclear Physics', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch2_P3.db
title:
the-mass-of-7-n-15-is-1500011-amu-mass-of-8-o-16-is-1599492-amu-and-m-p-100783-amu-determine-binding-energy-of-last-proton-of-8-o-16

slug:
the-mass-of-7-n-15-is-1500011-amu-mass-of-8-o-16-is-1599492-amu-and-m-p-100783-amu-determine-binding-energy-of-last-proton-of-8-o-16-92144

question:
<div class="question">The mass of $_{7} \mathrm{~N}^{15}$ is $15.00011 \mathrm{amu}$, mass of $_{8} \mathrm{O}^{16}$ is 15.99492 amu and $m_{p}=1.00783$ amu. Determine binding energy of last proton of ${ }_{8} \mathrm{O}^{16}$.</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2.13 \mathrm{MeV}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$0.13 \mathrm{MeV}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$10 \mathrm{MeV}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$12.13 \mathrm{MeV}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$12.13 \mathrm{MeV}$</span> </div>

answer:
<div class="solution">$\mathrm{M}\left({ }_{8} \mathrm{O}^{16}\right)=\mathrm{M}\left({ }_{7} \mathrm{~N}^{15}\right)+1 \mathrm{~m}_{\mathrm{p}}$
<br/> <br/>binding energy of last proton $=M\left(N^{15}\right)+m_{P}-M\left({ }_{1} O^{16}\right)$
<br/> <br/>$=15.00011+1.00783-15.99492$
<br/> <br/>$=0.01302 \mathrm{amu}=12.13 \mathrm{MeV}$</div>


tags: ['Chemistry', 'Electrochemistry', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P3.db
title:
the-mass-of-the-substance-deposited-when-one-faraday-of-charge-is-passed-through-its-solution-is-equal-to

slug:
the-mass-of-the-substance-deposited-when-one-faraday-of-charge-is-passed-through-its-solution-is-equal-to-13214

question:
<div class="question">The mass of the substance deposited when one Faraday of charge is passed through its solution is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">relative equivalent weight</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">gram equivalent weight</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">specific equivalent weight</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of the above</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">gram equivalent weight</span> </div>

answer:
<div class="solution">The mass of the substance deposited when one Faraday of charge is passed through its solution is equal to gram equivalent weight.</div>


tags: ['Physics', 'Laws of Motion', 'JEE Main']
source_db: MarksBatch2_P3.db
title:
the-masses-of-blocks-a-and-b-are-m-and-m-respectively-between-a-and-b-there-is-a-constant-frictional-force-f-and-b-can-slide-on-a-smooth-horizontal-su-1

slug:
the-masses-of-blocks-a-and-b-are-m-and-m-respectively-between-a-and-b-there-is-a-constant-frictional-force-f-and-b-can-slide-on-a-smooth-horizontal-su-1-64252

question:
<div class="question">The masses of blocks $\mathrm{A}$ and $\mathrm{B}$ are $\mathrm{m}$ and $\mathrm{M}$ respectively. Between $\mathrm{A}$ and $\mathrm{B}$, there is a constant frictional force $\mathrm{F}$ and $\mathrm{B}$ can slide on a smooth horizontal surface. A is set in motion with velocity while $\mathrm{B}$ is at rest. What is the distance moved by A relative to B before they move with the same velocity?
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/VAoKgexswxHluYeDnxnZeM2i7knd07V9PafTJJbX2gA.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\mathrm{mMv}_{0}^{2}}{\mathrm{~F}(\mathrm{~m}-\mathrm{M})}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\mathrm{mMv}_{0}^{2}}{2 \mathrm{~F}(\mathrm{~m}-\mathrm{M})}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\mathrm{mMv}_{0}^{2}}{\mathrm{~F}(\mathrm{~m}+\mathrm{M})}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{\mathrm{mMv}_{0}^{2}}{2 \mathrm{~F}(\mathrm{M}+\mathrm{m})}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\mathrm{mMv}_{0}^{2}}{2 \mathrm{~F}(\mathrm{M}+\mathrm{m})}$</span> </div>

answer:
<div class="solution">For the blocks A and B FBD as shown below
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/6ow0RGQaH5-5nRGRRQuYDbTuvgtP8lc3H-a5Z_EoIvY.original.fullsize.png"/><br/> <br/> <br/>Equations of motion
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>\mathrm{a}_{\mathrm{A}}=\frac{\mathrm{F}}{\mathrm{M}}(\text { in }-\mathrm{x} \text { direction }) \\
<br/> <br/>\mathrm{a}_{\mathrm{B}}=\frac{\mathrm{F}}{\mathrm{M}}(\text { in }+\mathrm{x} \text { direction) }
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>Relative acceleration, of A w.r.t. B,
<br/> <br/>$$
<br/> <br/>\begin{aligned}
<br/> <br/>\mathrm{a}_{\mathrm{A}, \mathrm{B}} &amp;=\mathrm{a}_{\mathrm{A}}-\mathrm{a}_{\mathrm{B}}=-\frac{\mathrm{F}}{\mathrm{m}}-\frac{\mathrm{F}}{\mathrm{M}} \\
<br/> <br/>&amp;=-\mathrm{F}\left(\frac{\mathrm{M}+\mathrm{m}}{\mathrm{Mm}}\right)(\text { along }-\mathrm{x} \text { direction })
<br/> <br/>\end{aligned}
<br/> <br/>$$
<br/> <br/>Initial relative velocity of A w.r.t. B, $\mathrm{u}_{\mathrm{AB}}=\mathrm{v}_{0}$
<br/> <br/>using equation $v^{2}=u^{2}+2$ as
<br/> <br/>$$
<br/> <br/>0=v_{0}^{2}-\frac{2 \mathrm{~F}(\mathrm{~m}+\mathrm{M}) \mathrm{S}}{\mathrm{Mm}} \Rightarrow \mathrm{S}=\frac{\mathrm{Mmv}_{0}^{2}}{2 \mathrm{~F}(\mathrm{~m}+\mathrm{M})}
<br/> <br/>$$
<br/> <br/>i.e., Distance moved by A relative to B
<br/> <br/>$$
<br/> <br/>\mathrm{S}_{\mathrm{AB}}=\frac{\mathrm{Mmv}_{0}^{2}}{2 \mathrm{~F}(\mathrm{~m}+\mathrm{M})}
<br/> <br/>$$</div>


tags: ['Physics', 'Laws of Motion', 'BITSAT', 'BITSAT 2015']
source_db: MarksBatch2_P3.db
title:
the-masses-of-blocks-a-and-b-are-m-and-m-respectively-between-a-and-b-there-is-a-constant-frictional-force-f-and-b-can-slide-on-a-smooth-horizontal-su

slug:
the-masses-of-blocks-a-and-b-are-m-and-m-respectively-between-a-and-b-there-is-a-constant-frictional-force-f-and-b-can-slide-on-a-smooth-horizontal-su-97290

question:
<div class="question">The masses of blocks $\mathrm{A}$ and $\mathrm{B}$ are $\mathrm{m}$ and $\mathrm{M}$ respectively. Between $\mathrm{A}$ and $\mathrm{B}$, there is a constant frictional force $\mathrm{F}$ and $\mathrm{B}$ can slide on a smooth horizontal surface. A is set in motion with velocity while $\mathrm{B}$ is at rest. What is the distance moved by A relative to B before they move with the same velocity?
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/VAoKgexswxHluYeDnxnZeM2i7knd07V9PafTJJbX2gA.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\mathrm{mMv}_{0}^{2}}{\mathrm{~F}(\mathrm{~m}-\mathrm{M})}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\mathrm{mMv}_{0}^{2}}{2 \mathrm{~F}(\mathrm{~m}-\mathrm{M})}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\mathrm{mMv}_{0}^{2}}{\mathrm{~F}(\mathrm{~m}+\mathrm{M})}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{\mathrm{mMv}_{0}^{2}}{2 \mathrm{~F}(\mathrm{M}+\mathrm{m})}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\mathrm{mMv}_{0}^{2}}{2 \mathrm{~F}(\mathrm{M}+\mathrm{m})}$</span> </div>

answer:
<div class="solution">For the blocks A and B FBD as shown below
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/6ow0RGQaH5-5nRGRRQuYDbTuvgtP8lc3H-a5Z_EoIvY.original.fullsize.png"/><br/> <br/> <br/>Equations of motion
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>\mathrm{a}_{\mathrm{A}}=\frac{\mathrm{F}}{\mathrm{M}}(\text { in }-\mathrm{x} \text { direction }) \\
<br/> <br/>\mathrm{a}_{\mathrm{B}}=\frac{\mathrm{F}}{\mathrm{M}}(\text { in }+\mathrm{x} \text { direction) }
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>Relative acceleration, of A w.r.t. B,
<br/> <br/>$$
<br/> <br/>\begin{aligned}
<br/> <br/>\mathrm{a}_{\mathrm{A}, \mathrm{B}} &amp;=\mathrm{a}_{\mathrm{A}}-\mathrm{a}_{\mathrm{B}}=-\frac{\mathrm{F}}{\mathrm{m}}-\frac{\mathrm{F}}{\mathrm{M}} \\
<br/> <br/>&amp;=-\mathrm{F}\left(\frac{\mathrm{M}+\mathrm{m}}{\mathrm{Mm}}\right)(\text { along }-\mathrm{x} \text { direction })
<br/> <br/>\end{aligned}
<br/> <br/>$$
<br/> <br/>Initial relative velocity of A w.r.t. B, $\mathrm{u}_{\mathrm{AB}}=\mathrm{v}_{0}$
<br/> <br/>using equation $v^{2}=u^{2}+2$ as
<br/> <br/>$$
<br/> <br/>0=v_{0}^{2}-\frac{2 \mathrm{~F}(\mathrm{~m}+\mathrm{M}) \mathrm{S}}{\mathrm{Mm}} \Rightarrow \mathrm{S}=\frac{\mathrm{Mmv}_{0}^{2}}{2 \mathrm{~F}(\mathrm{~m}+\mathrm{M})}
<br/> <br/>$$
<br/> <br/>i.e., Distance moved by A relative to B
<br/> <br/>$$
<br/> <br/>\mathrm{S}_{\mathrm{AB}}=\frac{\mathrm{Mmv}_{0}^{2}}{2 \mathrm{~F}(\mathrm{~m}+\mathrm{M})}
<br/> <br/>$$</div>


title: the-matrix-a24-a5-i-where-i-is-identity-matrix-and-aleftbeginarraycc1-2-4-3endarrayright-equals
tags: ['Mathematics', 'Matrices', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P3.db
slug:
the-matrix-a24-a5-i-where-i-is-identity-matrix-and-aleftbeginarraycc1-2-4-3endarrayright-equals-43455

question:
<div class="question">The matrix \(A^2+4 A-5 I\), where \(I\) is identity matrix and \(A=\left[\begin{array}{cc}1 &amp; 2 \\ 4 &amp; -3\end{array}\right]\), equals :</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\(4\left[\begin{array}{ll}2 &amp; 1 \\ 2 &amp; 0\end{array}\right]\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(4\left[\begin{array}{cc}0 &amp; -1 \\ 2 &amp; 2\end{array}\right]\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(32\left[\begin{array}{ll}2 &amp; 1 \\ 2 &amp; 0\end{array}\right]\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(32\left[\begin{array}{ll}1 &amp; 1 \\ 1 &amp; 0\end{array}\right]\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(4\left[\begin{array}{ll}2 &amp; 1 \\ 2 &amp; 0\end{array}\right]\)</span> </div>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; A^2+4 A-5 I=A \times A+4 A-5 I \\<br/>= &amp; {\left[\begin{array}{cc}<br/>1 &amp; 2 \\<br/>4 &amp; -3<br/>\end{array}\right] \times\left[\begin{array}{cc}<br/>1 &amp; 2 \\<br/>4 &amp; -3<br/>\end{array}\right]+4\left[\begin{array}{cc}<br/>1 &amp; 2 \\<br/>4 &amp; -3<br/>\end{array}\right]-5\left[\begin{array}{ll}<br/>1 &amp; 0 \\<br/>0 &amp; 1<br/>\end{array}\right] } \\<br/>= &amp; {\left[\begin{array}{cc}<br/>9 &amp; -4 \\<br/>-8 &amp; 17<br/>\end{array}\right]+\left[\begin{array}{cc}<br/>4 &amp; 8 \\<br/>16 &amp; -12<br/>\end{array}\right]-\left[\begin{array}{cc}<br/>5 &amp; 0 \\<br/>0 &amp; 5<br/>\end{array}\right] } \\<br/>= &amp; {\left[\begin{array}{cc}<br/>9+4-5 &amp; -4+8-0 \\<br/>-8+16-0 &amp; 17-12-5<br/>\end{array}\right]=\left[\begin{array}{cc}<br/>8 &amp; 4 \\<br/>8 &amp; 0<br/>\end{array}\right] } \\<br/>= &amp; 4\left[\begin{array}{cc}<br/>2 &amp; 1 \\<br/>2 &amp; 0<br/>\end{array}\right]<br/>\end{aligned}\)</div>


title: the-maximum-value-of-f-x-x-sin-2-x-x-0-2-is
slug: the-maximum-value-of-f-x-x-sin-2-x-x-0-2-is-87302
tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P3.db
question:
<div class="question">The maximum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>x</mi><mo>+</mo><mi>sin</mi><mn>2</mn><mi>x</mi><mo>,</mo><mi>x</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>2</mn><mi>π</mi><mo>]</mo></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>π</mi><mn>2</mn></mfrac></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>π</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mn>4</mn></mfrac></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>3</mn><mi>π</mi></mrow><mn>2</mn></mfrac></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>π</mi></math></span> </div>

answer:
<div class="solution"><p>Given, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>x</mi><mo>+</mo><mi>sin</mi><mn>2</mn><mi>x</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><msup><mi>f</mi><mo>'</mo></msup><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>1</mn><mo>+</mo><mn>2</mn><mi>cos</mi><mn>2</mn><mi>x</mi></math></p><p>For maximum or minimum value, <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mn>1</mn><mo>+</mo><mn>2</mn><mi>cos</mi><mn>2</mn><mi>x</mi><mo>=</mo><mn>0</mn><mo>⇒</mo><mi>cos</mi><mn>2</mn><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mn>2</mn><mi>x</mi><mo>=</mo><mn>2</mn><mi>n</mi><mi>π</mi><mo>±</mo><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mn>3</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>x</mi><mo>=</mo><mi>n</mi><mi>π</mi><mo>±</mo><mfrac><mi>π</mi><mn>3</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>x</mi><mo>=</mo><mfrac><mi>π</mi><mn>3</mn></mfrac><mo>,</mo><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mn>3</mn></mfrac><mo>,</mo><mfrac><mrow><mn>4</mn><mi>π</mi></mrow><mn>3</mn></mfrac><mo>,</mo><mfrac><mrow><mn>5</mn><mi>π</mi></mrow><mn>3</mn></mfrac></math></p><p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>,</mo><mo> </mo><mi>f</mi><mfenced><mfrac><mi>π</mi><mn>3</mn></mfrac></mfenced><mo>,</mo><mi>f</mi><mfenced><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mn>3</mn></mfrac></mfenced><mo>,</mo><mo> </mo><mi>f</mi><mfenced><mfrac><mrow><mn>4</mn><mi>π</mi></mrow><mn>3</mn></mfrac></mfenced><mo>,</mo><mo> </mo><mi>f</mi><mfenced><mfrac><mrow><mn>5</mn><mi>π</mi></mrow><mn>3</mn></mfrac></mfenced><mo>,</mo><mo> </mo><mi>f</mi><mo>(</mo><mn>2</mn><mi>π</mi><mo>)</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>0</mn><mo>,</mo><mi>f</mi><mfenced><mfrac><mi>π</mi><mn>3</mn></mfrac></mfenced><mo>=</mo><mfrac><mi>π</mi><mn>3</mn></mfrac><mo>+</mo><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac><mo>,</mo><mi>f</mi><mfenced><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mn>3</mn></mfrac></mfenced><mo>=</mo><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mn>3</mn></mfrac><mo>-</mo><mn>0</mn><mo>.</mo><mn>8</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>f</mi><mfenced><mfrac><mrow><mn>4</mn><mi>π</mi></mrow><mn>3</mn></mfrac></mfenced><mo>=</mo><mfrac><mrow><mn>4</mn><mi>π</mi></mrow><mn>3</mn></mfrac><mo>+</mo><mn>0</mn><mo>.</mo><mn>8</mn><mo>,</mo><mi>f</mi><mfenced><mfrac><mrow><mn>5</mn><mi>π</mi></mrow><mn>3</mn></mfrac></mfenced><mo>=</mo><mfrac><mrow><mn>5</mn><mi>π</mi></mrow><mn>3</mn></mfrac><mo>-</mo><mn>0</mn><mo>.</mo><mn>8</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>f</mi><mfenced><mrow><mn>2</mn><mi>π</mi></mrow></mfenced><mo>=</mo><mn>2</mn><mi>π</mi><mo>+</mo><mn>0</mn><mo>=</mo><mn>2</mn><mi>π</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> Maximum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>2</mn><mi>π</mi></math></p></div>


title: the-maximum-value-of-z-4-x-2-y-subject-to-constraints-2-x-3-y-18-x-y-10-and-x-y-0-is
slug: the-maximum-value-of-z-4-x-2-y-subject-to-constraints-2-x-3-y-18-x-y-10-and-x-y-0-is-14668
tags: ['Mathematics', 'Linear Programming', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P3.db
question:
<div class="question">The maximum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi><mo>=</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi></math> subject to constraints <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>≤</mo><mn>18</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>≥</mo><mn>10</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>,</mo><mi>y</mi><mo>≥</mo><mn>0</mn></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>36</mn></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>40</mn></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">None of these</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">We have,<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi><mo>=</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi></math></p><p>And,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>≤</mo><mn>18</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>≥</mo><mn>10</mn></math></p><p>Let</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>1</mn></msub><mo>:</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>18</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>2</mn></msub><mo>:</mo><mo> </mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mn>10</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>3</mn></msub><mo>:</mo><mi>x</mi><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>4</mn></msub><mo>:</mo><mi>y</mi><mo>=</mo><mn>0</mn></math></p><p>Then the feasible region is</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/3697c4e7-aefa-40a0-89bf-3ff4d8185590-image.png" style="width: 250px; height: 238px;"/></p><p>From the figure, it is clear that there is no common area. So, we cannot find maximum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Z</mi></math>.</p><p>Hence, option <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>d</mi></mfenced></math> is correct.</p></div>


title: the-maximum-value-of-z-5-x-2-y-subject-to-constraints-x-y-7-x-2-y-10-x-y-0
slug: the-maximum-value-of-z-5-x-2-y-subject-to-constraints-x-y-7-x-2-y-10-x-y-0-32714
tags: ['Mathematics', 'Linear Programming', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">35</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">The maximum value of $z=5 x+2 y$ subject to constraints $x+y \leq 7, x+2 y \leq 10, x, y \geq 0$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">10</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">26</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">35</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">70</span> </li> </ul>

answer:
<div class="solution">Change the inequalities into equations and draw the graph of lines, thus we get the required feasible region as shown below.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/PM-tPvPQ2yDHDSeIf6ZrVRVLLrMIduwNfTr1Y8ScXcU.original.fullsize.png"/><br/><br/>The region bounded by the vertices $A(0,5), B(4,3), C(7,0)$<br/>The objective function is maximum at $\mathrm{C}(7,0)$ and $\operatorname{Max} z=5 \times 7+2 \times 0=35$.</div>


title: the-maximum-value-of-z-6-x-8-y-subject-to-constraints-2-x-y-30-x-2-y-24-and-x-0-y-0-is
slug: the-maximum-value-of-z-6-x-8-y-subject-to-constraints-2-x-y-30-x-2-y-24-and-x-0-y-0-is-20806
tags: ['Mathematics', 'Linear Programming', 'BITSAT', 'BITSAT 2020']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">120</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">The maximum value of $z=6 x+8 y$ subject to constraints $2 x+y \leq 30, x+2 y \leq 24$ and $x \geq 0, y \geq 0$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">90</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">120</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">96</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">240</span> </li> </ul>

answer:
<div class="solution">Here, $2 x+y \leq 30, x+2 y \leq 24, x, y \geq 0$
<br/> <br/>The shaded region represents the feasible region, hence $\mathrm{z}=6 \mathrm{x}+8 \mathrm{y}$. Obviously it is maximum at (12,6) Hence $z=12 \times 6+8 \times 6=120$
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/qxz5bvmr2xmJCHqxpyt9F0AeSB9p03OvCuJniSYliOI.original.fullsize.png"/></div>


title: the-maximum-value-of-z-9-x-13-y-subject-to-constraints-2-x-3-y-18-2-x-y-10-x-0-y-0-is
slug: the-maximum-value-of-z-9-x-13-y-subject-to-constraints-2-x-3-y-18-2-x-y-10-x-0-y-0-is-63422
tags: ['Mathematics', 'Mathematical Induction', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P3.db
question:
<div class="question">The maximum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><mn>9</mn><mi>x</mi><mo>+</mo><mn>13</mn><mi>y</mi></math> subject to constraints <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>≤</mo><mn>18</mn><mo>,</mo><mn>2</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>≤</mo><mn>10</mn><mo>,</mo><mi>x</mi><mo>≥</mo><mn>0</mn><mo>,</mo><mi>y</mi><mo>≥</mo><mn>0</mn></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>130</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>81</mn></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>79</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>99</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>79</mn></math></span> </div>

answer:
<div class="solution"><blockquote><p>Given, constraints are<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>y</mi><mo>≤</mo><mn>18</mn><mo>,</mo><mn>2</mn><mi>x</mi><mo>+</mo><mi>y</mi><mo>≤</mo><mn>10</mn><mo>,</mo><mi>x</mi><mo>≥</mo><mn>0</mn><mo>,</mo><mi>y</mi><mo>≥</mo><mn>0</mn></math>.</p><p>The feasible region is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi><mi>A</mi><mi>B</mi><mi>C</mi><mi>O</mi></math>.</p><p>Also, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><mn>9</mn><mi>x</mi><mo>+</mo><mn>13</mn><mi>y</mi></math></p><p><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/c2f3790a-5526-4e70-a0ca-9837eb432ade-d62.png" style="width: 326px; height: 230px;"/></p><p>At <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo><mo>,</mo><mi>z</mi><mo>=</mo><mn>0</mn></math></p><p>At <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>(</mo><mn>5</mn><mo>,</mo><mn>0</mn><mo>)</mo><mo>,</mo><mi>z</mi><mo>=</mo><mn>45</mn></math></p><p>At <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>)</mo><mo>,</mo><mi>z</mi><mo>=</mo><mn>27</mn><mo>+</mo><mn>52</mn><mo>=</mo><mn>79</mn></math></p><p>At <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>(</mo><mn>0</mn><mo>,</mo><mn>6</mn><mo>)</mo><mo>,</mo><mi>z</mi><mo>=</mo><mn>78</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> Maximum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>79</mn></math>.</p></blockquote></div>


tags: ['Mathematics', 'Probability', 'BITSAT', 'BITSAT 2021']
source_db: MarksBatch2_P3.db
title:
the-mean-and-variance-of-a-random-variable-x-having-binomial-distribution-are-4-and-2-respectively-then-p-x-1-is-1

slug:
the-mean-and-variance-of-a-random-variable-x-having-binomial-distribution-are-4-and-2-respectively-then-p-x-1-is-1-96953

question:
<div class="question">The mean and variance of a random variable $\mathrm{X}$ having binomial distribution are 4 and 2 respectively, then $\mathrm{P}(\mathrm{X}=1)$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{4}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{1}{32}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{16}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{8}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{32}$</span> </div>

answer:
<div class="solution">$\begin{aligned} n p &amp;=4 \\ n p q &amp;=2 \\ \Rightarrow q &amp;=\frac{1}{2}, p=\frac{1}{2}, n=8 \\ \mathrm{P}(\mathrm{X}&amp;=1)={ }^{8} \mathrm{C}_{1}\left(\frac{1}{2}\right)\left(\frac{1}{2}\right)^{7}=8 \cdot \frac{1}{2^{8}}=\frac{1}{2^{5}}=\frac{1}{32} \end{aligned}$</div>


tags: ['Mathematics', 'Statistics', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P3.db
title:
the-mean-of-five-observations-is-4-and-their-variance-is-52-if-three-of-these-observations-are-1-2-and-6-then-the-other-two-are

slug:
the-mean-of-five-observations-is-4-and-their-variance-is-52-if-three-of-these-observations-are-1-2-and-6-then-the-other-two-are-41678

question:
<div class="question">The mean of five observations is $4$ and their variance is $5.2$. If three of these observations are $1$,$2$ and $6$ , then the other two are</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>$2$ and $9$
<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <br/>$3$ and $8$
<br/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"> <br/>$4$ and $7$
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>$5$ and $6$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>$4$ and $7$
<br/></span> </div>

answer:
<div class="solution">Let the two unknown items be $x$ and $y$. Then,
<br/>$\operatorname{Mean}=4 \Rightarrow \frac{1+2+6+x+y}{5}=4$
<br/>$\Rightarrow \quad x+y=11\dots(i)$
<br/>and variance $=5.2$
<br/>$\Rightarrow \frac{1^2+2^2+6^2+x^2+y^2}{5}-(\text { mean })^2=5.2$
<br/>$\Rightarrow \quad 41+x^2+y^2=5(5.2+16)$
<br/>$\begin{array}{ll}\Rightarrow &amp; 41+x^2+y^2=106 \\ \Rightarrow &amp; x^2+y^2=65\end{array}$
<br/>On solving Eqs. (i) and (ii), we get
<br/>$x=4, y=7 \text { or } x=7, y=4$</div>


tags: ['Mathematics', 'Statistics', 'BITSAT', 'BITSAT 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: MarksBatch2_P3.db
title:
the-mean-square-deviation-of-a-set-of-n-observation-x-1-x-2-2-x-n-about-a-point-c-is-defined-as-n-1-i-1-n-x-i-c-2-the-mean-square-deviations-about-2-a

slug:
the-mean-square-deviation-of-a-set-of-n-observation-x-1-x-2-2-x-n-about-a-point-c-is-defined-as-n-1-i-1-n-x-i-c-2-the-mean-square-deviations-about-2-a-33914

question:
<div class="question">The mean square deviation of a set of n observation $\mathrm{x}_{1}, \mathrm{x}_{2^{2}}, \ldots \mathrm{x}_{\mathrm{n}}$ about a point $\mathrm{c}$ is defined as $\frac{1}{n} \sum_{i=1}^{n}\left(x_{i}-c\right)^{2}$
<br/> <br/>The mean square deviations about -2 and 2 are 18 and 10 respectively, the standard deviation of this set of observations is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">3</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

answer:
<div class="solution">We have $\frac{1}{n} \sum_{i=1}^{n}\left(x_{i}+2\right)^{2}=18$ and
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>\frac{1}{n} \sum_{i=1}^{n}\left(x_{i}-2\right)^{2}=10 \\
<br/> <br/>\Rightarrow \sum_{i=1}^{n}\left(x_{i}+2\right)^{2}=18 n \text { and }
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>\sum_{i=1}^{n}\left(x_{i}-2\right)^{2}=10 n \\
<br/> <br/>\Rightarrow \sum_{i=1}^{n}\left(x_{i}+2\right)^{2}+\sum_{i=1}^{n}\left(x_{i}-2\right)^{2}=28 n
<br/> <br/>\end{array}
<br/> <br/>$$
<br/> <br/>and $\sum_{i=1}^{n}\left(x_{i}+2\right)^{2}-\sum_{i=1}^{n}\left(x_{i}-2\right)^{2}=8 n$
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>\Rightarrow 2 \sum_{i=1}^{n}\left(x_{i}+4\right)^{2}=28 n \text { and } 2 \sum_{i=1}^{n} 4 x_{i}=8 n \\
<br/> <br/>\Rightarrow \sum_{i=1}^{n} x_{1}^{2}+4 n=14 n \text { and } \sum_{i=1}^{n} x_{i}=n \\
<br/> <br/>\Rightarrow \sum_{i=1}^{n} x_{i}^{2}=10 n \text { and } \sum_{i=1}^{n} x_{i}=n \\
<br/> <br/>\therefore \sigma=\sqrt{\frac{1}{n} \sum_{i=1}^{n} x_{i}^{2}-\left(\frac{1}{n} \sum_{i=1^{\prime}}^{n} x_{i}\right)^{2}}=\sqrt{\frac{10 n}{n}-\left(\frac{n}{n}\right)^{2}}=3
<br/> <br/>\end{array}
<br/> <br/>$$</div>


title: the-metal-present-in-vitamin-b-12-is
slug: the-metal-present-in-vitamin-b-12-is-73941
question: <div class="question">The metal present in vitamin $\mathrm{B}_{12}$ is</div>
tags: ['Chemistry', 'Biomolecules', 'BITSAT', 'BITSAT 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">cobalt</span> </div>
answer: <div class="solution">Cobalt is present in vitamin $\mathrm{B}_{12}$.</div>
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">magnesium</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">cobalt</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">copper</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">zinc</span> </li> </ul>


title: the-middle-term-in-the-expansion-of-x-10-10-x-10-is
slug: the-middle-term-in-the-expansion-of-x-10-10-x-10-is-21205
tags: ['Mathematics', 'Binomial Theorem', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">The middle term in the expansion of $\left(\frac{10}{x}+\frac{x}{10}\right)^{10}$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">${ }^{10} \mathrm{C}_5$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">${ }^{10} \mathrm{C}_6$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> ${ }^{10} \mathrm{C}_5 \frac{1}{x^{10}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">${ }^{10} \mathrm{C}_5 \mathrm{x}^{10}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">${ }^{10} \mathrm{C}_5$</span> </div>

answer:
<div class="solution">General term $=T_{r+1}$<br/>$={ }^{10} \mathrm{C}_r\left(\frac{10}{x}\right)^{10-r}\left(\frac{x}{10}\right)^r$<br/>Here $n=10$, which is an even number.<br/>Now, $\left[\frac{10}{2}+1\right]^{\text {th }}$ term i.e. $6^{\text {th }}$ term is the middle term.<br/>Hence, middle term $=T_6$<br/>$\mathrm{T}_{5+1}={ }^{10} \mathrm{C}_5\left(\frac{10}{x}\right)^{10-5}\left(\frac{x}{10}\right)^5$<br/>$={ }^{10} \mathrm{C}_5\left(\frac{10}{x}\right)^5\left(\frac{x}{10}\right)^5={ }^{10} \mathrm{C}_5$.</div>


title: the-minimum-value-of-fracx4y4z2x-y-z-for-positive-real-number-x-y-z-is
slug: the-minimum-value-of-fracx4y4z2x-y-z-for-positive-real-number-x-y-z-is-63170
tags: ['Mathematics', 'Limits', 'BITSAT', 'BITSAT 2010']
source_db: MarksBatch2_P3.db
question:
<div class="question">The minimum value of \(\frac{x^4+y^4+z^2}{x y z}\) for positive real number \(x, y, z\) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\sqrt{2}\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(2 \sqrt{2}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(4 \sqrt{2}\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(8 \sqrt{2}\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(2 \sqrt{2}\)</span> </div>

answer:
<div class="solution">By A.M. \(\geq\) G.M.<br/>\(x^4+y^4 \geq 2 x^2 y^2\) and \(2 x^2 y^2+z^2 \geq \sqrt{8} x y z\).<br/>\(\Rightarrow \frac{x^4+y^4+z^2}{x y z} \geq \sqrt{8}\)</div>


title: the-minimum-value-of-fxsin-4-xcos-4-x-in-the-interval-left0-fracpi2right-is
slug: the-minimum-value-of-fxsin-4-xcos-4-x-in-the-interval-left0-fracpi2right-is-39948
tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P3.db
question:
<div class="question">The minimum value of \(f(x)=\sin ^4 x+\cos ^4 x\) in the interval \(\left(0, \frac{\pi}{2}\right)\) is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\(\frac{1}{2}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(\sqrt{2}\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\frac{1}{2}\)</span> </div>

answer:
<div class="solution">Let \(y=\sin ^4 x+\cos ^4 x\)<br/>\(\begin{aligned}<br/>&amp; \frac{d y}{d x}=4 \sin ^3 x \cos x+4 \cos ^3 x(-\sin x) \\<br/>&amp; =4 \sin x \operatorname{cox}\left(\sin ^2 x-\cos ^2 x\right) \\<br/>&amp; =(2 \sin 2 x)(-\cos 2 x)=-\sin 4 x \\<br/>&amp; \therefore \frac{d y}{d x}=0 \Rightarrow \sin 4 x=0 \Rightarrow 4 x=0, \pi, 2 \pi, 3 \pi<br/>\end{aligned}\)<br/>or \(\mathrm{x}=0, \frac{\pi}{4}, \frac{\pi}{2}, \frac{3 \pi}{4}, \ldots \ldots \ldots, \Rightarrow \mathrm{x}=\frac{\pi}{4}\)</div>


title: the-minimum-value-of-the-function-yx42-x21-in-the-interval-leftfrac12-2right-is
slug: the-minimum-value-of-the-function-yx42-x21-in-the-interval-leftfrac12-2right-is-86192
tags: ['Mathematics', 'Application of Derivatives', 'BITSAT', 'BITSAT 2009']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">The minimum value of the function \(y=x^4-2 x^2+1\) in the interval \(\left[\frac{1}{2}, 2\right]\) is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">8</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">9</span> </li> </ul>

answer:
<div class="solution">\(\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x}^4-2 \mathrm{x}^2+1\right)=4 \mathrm{x}\left(\mathrm{x}^2-1\right)\)<br/>For \(\max\). or \(\min , \frac{\mathrm{dy}}{\mathrm{dx}}=0\)<br/>\(4 \mathrm{x}\left(\mathrm{x}^2-1\right)=0 ;\), either \(\mathrm{x}=0\) or \(\mathrm{x}= \pm 1\)<br/>\(x=0\) and \(x=-1\) does not belong to \(\left[\frac{1}{2}, 2\right]\)<br/>\(\frac{d^2 y}{d x^2}=12 x^2-4 \therefore\left(\frac{d^2 y}{d x^2}\right)_{x=1}\)<br/>\(=12(1)^2-4=8 &gt; 0\)<br/>\(\therefore \quad\) there is minimum value of function at \(\mathrm{x}=1\)<br/>\(\therefore \quad\) minimum value is<br/>\(y(1)=1^4-2(1)^2+1=1-2+1=0\)</div>


tags: ['Mathematics', 'Sequences and Series', 'BITSAT', 'BITSAT 2009']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">220</span> </div>
answer: <div class="solution">No Solution Available</div>
source_db: MarksBatch2_P3.db
title:
the-minimum-value-of-the-function-z4-x3-y-subject-to-the-constraints-3-x2-y-geq-1605-x2-y-geq-200-x2-y-geq-80-x-geq-0-y-geq-0-is

slug:
the-minimum-value-of-the-function-z4-x3-y-subject-to-the-constraints-3-x2-y-geq-1605-x2-y-geq-200-x2-y-geq-80-x-geq-0-y-geq-0-is-26661

question:
<div class="question">The minimum value of the function \(z=4 x+3 y\) subject to the constraints \(3 x+2 y \geq 160,5 x+2 y \geq 200\), \(x+2 y \geq 80, x \geq 0, y \geq 0\) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">320</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">300</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">220</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">200</span> </li> </ul>


tags: ['Physics', 'Laws of Motion', 'BITSAT', 'BITSAT 2013']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">30</span> </div>
source_db: MarksBatch2_P3.db
title:
the-minimum-velocity-in-ms-1-with-which-a-car-driver-must-traverse-a-flat-curve-of-radius-150-m-and-coefficient-of-friction-06-to-avoid-skidding-is-1

slug:
the-minimum-velocity-in-ms-1-with-which-a-car-driver-must-traverse-a-flat-curve-of-radius-150-m-and-coefficient-of-friction-06-to-avoid-skidding-is-1-92234

question:
<div class="question">The minimum velocity (in $\mathrm{ms}^{-1}$ ) with which a car driver must traverse a flat curve of radius $150 \mathrm{~m}$ and coefficient of friction 0.6 to avoid skidding is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">60</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">30</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">15</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">25</span> </li> </ul>

answer:
<div class="solution">The condition to avoid skidding,
<br/> <br/>$$
<br/> <br/>\mathrm{v}=\sqrt{\mu \mathrm{rg}}=\sqrt{0.6 \times 150 \times 10}=30 \mathrm{~m} / \mathrm{s}
<br/> <br/>$$</div>


title: the-modulus-of-3-i-1-i-3-2-2-i-is
slug: the-modulus-of-3-i-1-i-3-2-2-i-is-94263
question: <div class="question">The modulus of $\frac{(1+i \sqrt{3})(2+2 i)}{(\sqrt{3}-i)}$ is</div>
tags: ['Mathematics', 'Complex Number', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>2</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>4</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$3 \sqrt{2}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$2 \sqrt{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2 \sqrt{2}$</span> </div>

answer:
<div class="solution"><br/>$\begin{aligned} &amp; \frac{(1+i \sqrt{3})(2+2 i)}{\sqrt{3}-i}=\frac{2+2 \sqrt{3} i+2 i-2 \sqrt{3}}{\sqrt{3}-i} \\ &amp; =\frac{(2-2 \sqrt{3})+(2 \sqrt{3}+2) i}{\sqrt{3}-i} \times \frac{\sqrt{3}+i}{\sqrt{3}+i} \\ &amp; =\frac{2 \sqrt{3}-6+2 i-2 \sqrt{3} i+6 i+2 \sqrt{3} i-2 \sqrt{3}-2}{3+1} \\ &amp; =\frac{8 i-8}{4}=-2+2 i \\ &amp; \therefore \text { Modulus }=\sqrt{(-2)^2+(2)^2}=2 \sqrt{2}\end{aligned}$</div>


tags: ['Physics', 'Kinetic Theory of Gases', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch2_P3.db
title:
the-molecules-of-a-given-mass-of-a-gas-have-rms-velocity-of-200-ms-1-at-2-7-c-and-10-1-0-5-nm-2-pressure-when-the-temperature-and-pressure-of-the-gas-

slug:
the-molecules-of-a-given-mass-of-a-gas-have-rms-velocity-of-200-ms-1-at-2-7-c-and-10-1-0-5-nm-2-pressure-when-the-temperature-and-pressure-of-the-gas-48805

question:
<div class="question">The molecules of a given mass of a gas have
<br/> <br/>r.m.s. velocity of $200 \mathrm{~ms}^{-1}$ at $27^{\circ} \mathrm{C}$ and $1.0 \times 10^{5}$ $\mathrm{Nm}^{-2}$ pressure. When the temperature and pressure of the gas are respectively, $127^{\circ} \mathrm{C}$ and $0.05 \times 10^{5} \mathrm{Nm}^{-2},$ the r.m.s. velocity of its molecules in $\mathrm{ms}^{-1}$ is :</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$100 \sqrt{2}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{400}{\sqrt{3}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{100 \sqrt{2}}{3}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{100}{3}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{400}{\sqrt{3}}$</span> </div>

answer:
<div class="solution">Here $v_{1}=200 \mathrm{~m} / \mathrm{s}$; temperature $\mathrm{T}_{1}=27^{\circ} \mathrm{C}=27+273=300 \mathrm{k}$
<br/> <br/>temperature $\mathrm{T}_{2}=127^{\circ} \mathrm{C}=127+273=400 \mathrm{k}$
<br/> <br/>$\mathrm{V}=?$
<br/> <br/>R.M.S. Velocity, $\mathrm{V} \propto \sqrt{\mathrm{T}}$
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>\Rightarrow \quad \frac{v}{200}=\sqrt{\frac{400}{300}} \\
<br/> <br/>\Rightarrow \quad v=\frac{200 \times 2}{\sqrt{3}} \mathrm{~m} / \mathrm{s} \\
<br/> <br/>\Rightarrow \quad \mathrm{v}=\frac{400}{\sqrt{3}} \mathrm{~m} / \mathrm{s}
<br/> <br/>\end{array}
<br/> <br/>$$</div>


tags: ['Physics', 'Kinetic Theory of Gases', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P3.db
title:
the-molecules-of-a-given-mass-of-gas-have-a-root-mean-square-velocity-of-200-m-s-1-at-2-7-c-and-10-1-0-5-nm-2-pressure-when-the-temperature-is-12-7-c-

slug:
the-molecules-of-a-given-mass-of-gas-have-a-root-mean-square-velocity-of-200-m-s-1-at-2-7-c-and-10-1-0-5-nm-2-pressure-when-the-temperature-is-12-7-c-77761

question:
<div class="question">The molecules of a given mass of gas have a root mean square velocity of $200 \mathrm{~m} \mathrm{~s}^{-1}$ at $27^{\circ} \mathrm{C}$ and $1.0 \times 10^{5} \mathrm{Nm}^{-2}$ pressure. When the temperature is $127^{\circ} \mathrm{C}$ and the pressure $0.5 \times 10^{5} \mathrm{Nm}^{-2},$ the root mean square velocity in $\mathrm{ms}^{-1}$, is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{400}{\sqrt{3}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$100 \sqrt{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{100 \sqrt{2}}{3}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{100}{3}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{400}{\sqrt{3}}$</span> </div>

answer:
<div class="solution">$\frac{c_{2}}{c_{1}}=\sqrt{\frac{400}{300}}=\frac{2}{\sqrt{3}}$<br/>$$<br/>\Rightarrow c_{2}=\frac{2}{\sqrt{3}} \times 200=\frac{400}{\sqrt{3}} \mathrm{~ms}^{-1}<br/>$$</div>


title: the-most-acidic-oxide-for-nitrogen-is
slug: the-most-acidic-oxide-for-nitrogen-is-86800
question: <div class="question">The most acidic oxide for nitrogen is</div>
tags: ['Chemistry', 'p Block Elements (Group 15, 16, 17 & 18)', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>NO</mi><mn>2</mn></msub></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">N</mi><mn>2</mn></msub><mi mathvariant="normal">O</mi></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>NO</mi></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">N</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mn>5</mn></msub></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">N</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mn>5</mn></msub></math></span> </div>

answer:
<div class="solution"><p>The acidic nature of the oxides of nitrogen can be determined by the strength of the acid formed by the nitrogen oxide in water. </p><p>More higher be the <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mo>+</mo><mo>)</mo><mo> </mo><mi>ve</mi></math> oxidation number of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">N</mi></math>-atom in its oxide more acidic be the nature of oxide. Thus <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">N</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mn>5</mn></msub></math>, is most acidic oxide.</p></div>


tags: ['Chemistry', 'Aldehydes and Ketones', 'BITSAT', 'BITSAT 2016']
source_db: MarksBatch2_P3.db
title:
the-most-likely-acidcatalysed-aldol-condensation-products-of-each-of-the-two-aldehydes-i-and-ii-will-respectively-be

slug:
the-most-likely-acidcatalysed-aldol-condensation-products-of-each-of-the-two-aldehydes-i-and-ii-will-respectively-be-19349

question:
<div class="question">The most likely acid-catalysed aldol condensation products of each of the two aldehydes I and II will respectively be
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/By4AwB5W_mGFuI9f27n-3oWEnBWLlWOoVC5TO5NocBg.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/YfzXm1W701Gl2v68EZmQNoa0fFu8djAVJfo4eSBjQ4o.original.fullsize.png"/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/Ylzys93N-ZSIRXbt1vylEOJL6Odf_phz8Uw9RUTDGYg.original.fullsize.png"/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/PoAvGHp0a4eekqjkrxtecBC4JW_pH6MElPeh-QfW7-k.original.fullsize.png"/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/CXrj5IxqKLynmbWIPQuD_xB-TTEB18_PLzw5Nre5QSw.original.fullsize.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/CXrj5IxqKLynmbWIPQuD_xB-TTEB18_PLzw5Nre5QSw.original.fullsize.png"/></span> </div>

answer:
<div class="solution">\(\left[F e F_{6}\right]^{3-}\) complex ion possesses sp \(^{3} d^{2}\) hybridization.
<br/> <br/>The electronic configurations of \(C r, F e, C o\) and \(T i\) in the complexes are \((3 d)^{3},(3 d)^{5},(3 d)^{6}\) and \((3 d)^{1}\), respectively.
<br/> <br/>These involve \(d^{2} s p^{3}, s p^{3} d^{2}, d^{2} s p^{3}\) and \(d^{2} s p^{3}\) hybridization respectively.</div>


title: the-most-reactive-amine-towards-reactions-with-dil-hcl-is
slug: the-most-reactive-amine-towards-reactions-with-dil-hcl-is-35208
tags: ['Chemistry', 'Amines', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P3.db
question:
<div class="question">The most reactive amine towards reactions with dil. <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>HCl</mi></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>CH</mi><mn>3</mn></msub><mo>-</mo><msub><mi>NH</mi><mn>2</mn></msub></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mo>(</mo><msub><mi>CH</mi><mn>3</mn></msub><mo>)</mo></mrow><mn>2</mn></msub><mi>NH</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mfenced><msub><mi>CH</mi><mn>3</mn></msub></mfenced><mn>3</mn></msub><mo>·</mo><mi mathvariant="normal">N</mi></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>PhNH</mi><mn>2</mn></msub></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mo>(</mo><msub><mi>CH</mi><mn>3</mn></msub><mo>)</mo></mrow><mn>2</mn></msub><mi>NH</mi></math></span> </div>

answer:
<div class="solution"><p>More be the ability of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">N</mi></math>-atom of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><msub><mi>NH</mi><mn>2</mn></msub></math> group to donate lone pair of electrons to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>HCl</mi></math> (dil.), more be the reactivity of amine.</p><p>In (B) two alkyl groups are attached to the <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><msub><mi>NH</mi><mn>2</mn></msub></math> (having electron releasing nature) which increases the electron donating ability of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">N</mi></math>-atom of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><msub><mi>NH</mi><mn>2</mn></msub></math>. In aqueous medium, secondary amine is most basic. Hence, it is most reactive.  In (D), due to resonance, lone pair of electrons get delocalised and its basic strength is very less.</p></div>


title: the-most-stable-carbanion-among-the-following-is
slug: the-most-stable-carbanion-among-the-following-is-50338
question: <div class="question">The most stable carbanion among the following is:</div>
tags: ['Chemistry', 'General Organic Chemistry', 'BITSAT', 'BITSAT 2013']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/eIWE5DJ0bxoLGiid-9a4zGFAh6W02GG3tmecQNQkatY.original.fullsize.png"/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/5XQX2jfJbtTObtplxqseMDGrUe9kkjDZnNpKpFfx3q0.original.fullsize.png"/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/9y6QxRWo1h3SiaAt5E9oIhPpnevM9qQRQskE-NOSW20.original.fullsize.png"/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/MQDfpIkroNJDd_2YgJHnopdfFe2tWnFOWA4_-xY-Cb4.original.fullsize.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/MQDfpIkroNJDd_2YgJHnopdfFe2tWnFOWA4_-xY-Cb4.original.fullsize.png"/></span> </div>

answer:
<div class="solution">$-\mathrm{NO}_{2}$ group, being strong electron withdrawing, disperses the -ve charge, hence stabilizes the concerned carbanion.</div>


title: the-most-stable-ion-is-2
slug: the-most-stable-ion-is-2-97638
question: <div class="question">The most stable ion is</div>
tags: ['Chemistry', 'Coordination Compounds', 'BITSAT', 'BITSAT 2011']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\left[\mathrm{Fe}(\mathrm{OH})_{3}\right]^{3-}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\left[\mathrm{FeCl}_{6}\right]^{3-}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]^{3-}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left[\mathrm{Fe}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{3+}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left[\mathrm{FeCl}_{6}\right]^{3-}$</span> </div>

answer:
<div class="solution">A more basic ligand forms stable bond with metal ion, $\mathrm{Cl}^{-}$ is most basic amongst all.</div>


title: the-n-th-roots-of-unity-are-in
slug: the-n-th-roots-of-unity-are-in-58491
tags: ['Mathematics', 'Complex Number', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch2_P3.db
question:
<div class="question">The <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>n</mi><mi>th</mi></msup></math> roots of unity are in</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>AP</mi></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>GP</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>HP</mi></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>GP</mi></math></span> </div>

answer:
<div class="solution"><p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>cos</mi><mn>0</mn><mo>°</mo><mo>+</mo><mi>i</mi><mi>sin</mi><mn>0</mn><mo>°</mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><msup><mi>x</mi><mrow><mn>1</mn><mo>/</mo><mi>n</mi></mrow></msup><mo>=</mo><msup><mi>e</mi><mrow><mi>i</mi><mfenced><mrow><mn>2</mn><mi>π</mi><mi>r</mi></mrow></mfenced><mo>/</mo><mi>n</mi></mrow></msup><mo>;</mo><mo> </mo><mi>r</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>1</mn><mo>,</mo><mo> </mo><mn>2</mn><mo>,</mo><mo>…</mo></math></p><p>Then, the roots are <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>,</mo><mo> </mo><msup><mi>e</mi><mfrac><mrow><mn>2</mn><mi>π</mi><mi>i</mi></mrow><mi>n</mi></mfrac></msup><mo>,</mo><mo> </mo><msup><mi>e</mi><mfrac><mrow><mn>4</mn><mi>π</mi><mi>i</mi></mrow><mi>n</mi></mfrac></msup><mo>,</mo><mo>…</mo></math> which are clearly on <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>GP</mi></math> with common ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>e</mi><mrow><mn>2</mn><mi>π</mi><mi>i</mi><mo>/</mo><mi>n</mi></mrow></msup></math></p></div>


title: the-nearest-point-on-the-line-3-x-4-y-12-from-the-origin-is
slug: the-nearest-point-on-the-line-3-x-4-y-12-from-the-origin-is-48355
question: <div class="question">The nearest point on the line $3 x+4 y=12$ from the origin is</div>
tags: ['Mathematics', 'Straight Lines', 'BITSAT', 'BITSAT 2012']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\left(\frac{36}{25}, \frac{48}{25}\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\left(3, \frac{3}{4}\right)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\left(2, \frac{3}{2}\right)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(\frac{36}{25}, \frac{48}{25}\right)$</span> </div>

answer:
<div class="solution">If ' $\mathrm{D}$ ' be the foot of altitude, drawn from origin to the given line, then ${ }^{4} \mathrm{D}^{\prime}$ is the required point. Let $\angle \mathrm{OBA}=\theta$
<br/>$\Rightarrow \tan \theta=4 / 3$
<br/>$\Rightarrow \angle \mathrm{DOA}=\theta$
<br/>$\begin{aligned}
<br/>&amp;\text { we have } \quad \mathrm{OD}=12 / 5 \text { . }\\
<br/>&amp;\text { If } D \text { is }(h, k) \text { then } h=O D \cos \theta, k=O D \sin \theta\\
<br/>&amp;\Rightarrow \mathrm{h}=36 / 25, \mathrm{k}=48 / 25 .
<br/>\end{aligned}$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/5HcYlB5S4A4eT7SwEFhioTiafNfqqvnEdBOdZVh3PLM.original.fullsize.png"/></div>


title: the-negation-of-the-boolean-expression-q-p-p-q-is-logically-equivalent-to-1
slug: the-negation-of-the-boolean-expression-q-p-p-q-is-logically-equivalent-to-1-66376
tags: ['Mathematics', 'Mathematical Reasoning', 'BITSAT', 'BITSAT 2023 (Memory Based Paper 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">The negation of the Boolean expression $((\sim q)$ $\wedge p) \Rightarrow((\sim p) \vee q)$ is logically equivalent to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$p \Rightarrow q$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$q \Rightarrow p$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\sim(p \Rightarrow q)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\sim(q \Rightarrow p)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\sim(p \Rightarrow q)$</span> </div>

answer:
<div class="solution">Given boolen expression are<br/>$\begin{aligned}<br/>&amp; \sim p \vee q \equiv p \rightarrow q \text { and } \sim q \wedge p \equiv \sim(p \rightarrow q) \\<br/>&amp; \text { Negation of } \sim(p \rightarrow q) \rightarrow(p \rightarrow q) \\<br/>&amp; \text { is } \sim(p \rightarrow q) \wedge(\sim(p \rightarrow q)) \text { i.e., } \sim(p \rightarrow q)<br/>\end{aligned}$</div>


title: the-nightblindness-is-developed-due-to-deficiency-of-vitamin
slug: the-nightblindness-is-developed-due-to-deficiency-of-vitamin-82585
question: <div class="question">The night-blindness is developed due to deficiency of vitamin</div>
tags: ['Chemistry', 'Biomolecules', 'BITSAT', 'BITSAT 2017']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">B</mi><mn>6</mn></msub></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">C</mi></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">B</mi><mn>12</mn></msub></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">A</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">A</mi></math></span> </div>

answer:
<div class="solution"><p>Night blindness is developed due to deficiency of vitamin-<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">A</mi></math>.</p><p>Vitamin A deficiency leads to a disease known as night blindness. People with night blindness have a normal vision when adequate light is present but poor vision in the dark.</p><p>Vitamin A is necessary for the production of retinal, a component of rhodopsin. </p></div>


tags: ['Chemistry', 'Solid State', 'BITSAT', 'BITSAT 2016']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">18</span> </div>
source_db: MarksBatch2_P3.db
title:
the-non-stoichiometric-compound-fe-094-o-is-formed-when-x-of-fe-2-ions-are-replaced-by-as-many-3-2-fe-3-ions-x-is

slug:
the-non-stoichiometric-compound-fe-094-o-is-formed-when-x-of-fe-2-ions-are-replaced-by-as-many-3-2-fe-3-ions-x-is-40839

question:
<div class="question">The non stoichiometric compound $\mathrm{Fe}_{0.94} \mathrm{O}$ is formed when $\mathrm{x} \%$ of $\mathrm{Fe}^{2+}$ ions are replaced by as many $\frac{2}{3} \mathrm{Fe}^{3+}$ ions, $\mathrm{x}$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">18</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">12</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">15</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">6</span> </li> </ul>

answer:
<div class="solution">The number of $\mathrm{Fe}^{3+}$ ions replacing $\mathrm{x} \mathrm{Fe}^{2+}$ ions $=\frac{2 \mathrm{x}}{3}$ vacancies of cations
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>=x-\frac{2 x}{3}=x / 3 \\
<br/> <br/>\text { But } x / 3=1-0.94=0.06, x=0.06 \times 3 \\
<br/> <br/>=0.18=18 \%
<br/> <br/>\end{array}
<br/> <br/>$$</div>


title: the-normality-of-26-wtvol-solution-of-ammonia-density-0855-is-approximately
slug: the-normality-of-26-wtvol-solution-of-ammonia-density-0855-is-approximately-13269
tags: ['Chemistry', 'Some Basic Concepts of Chemistry', 'BITSAT', 'BITSAT 2016']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">15.3</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">The normality of $26 \%$ (wt/vol) solution of ammonia (density $=0.855$ ) is approximately:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1.5</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">0.4</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">15.3</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">4</span> </li> </ul>

answer:
<div class="solution">Wt. of $\mathrm{NH}_{3}=26 \mathrm{~g}=\frac{26}{17} \mathrm{~g} \mathrm{eq}=1.53 \mathrm{~g} \mathrm{eq}$
<br/> <br/>Vol. of soln. $=100 \mathrm{mc}=0.1 \mathrm{~L}$
<br/> <br/>$\therefore$ Normality $=\frac{1.53}{0.1}=15.3 \mathrm{~N}$</div>


title: the-number-of-3-c-2-e-bonds-present-in-diborane-is
slug: the-number-of-3-c-2-e-bonds-present-in-diborane-is-23055
question: <div class="question">The number of $3 \mathrm{C}-2 e^{-}$bonds present in diborane is</div>
tags: ['Chemistry', 'p Block Elements (Group 13 & 14)', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"> <br/>1
<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"> <br/>2
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>3
<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>4</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>2
<br/></span> </div>

answer:
<div class="solution">The structure of diborane is
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/-4EAmcvZnHO8X3GVHSqcsEvyeQS8-dTaftJ_-g9aGeo.original.fullsize.png"/><br/> <br/>In this structure, there are two $3 \mathrm{C}-2 e^{-}$bonds. 3 atoms, $\quad \mathrm{B}-\mathrm{H}-\mathrm{B}$ share two electrons and form an angular geometry, leading bent bond, also known as banana bond.</div>


title: the-number-of-all-three-elements-subsets-of-the-set-a-1-a-2-a-3-a-n-which-contain-a-3-is
slug: the-number-of-all-three-elements-subsets-of-the-set-a-1-a-2-a-3-a-n-which-contain-a-3-is-41623
tags: ['Mathematics', 'Permutation Combination', 'BITSAT', 'BITSAT 2014']
source_db: MarksBatch2_P3.db
question:
<div class="question">The number of all three elements subsets of the set $\left\{a_{1}, a_{2}, a_{3} \ldots a_{n}\right\}$ which contain $a_{3}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">${ }^{\mathrm{n}} \mathrm{C}_{3}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">${ }^{\mathrm{n}-1} \mathrm{C}_{3}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">${ }^{\mathrm{n}-1} \mathrm{C}_{2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">${ }^{\mathrm{n}-1} \mathrm{C}_{2}$</span> </div>

answer:
<div class="solution">The number of three elements subsets containing $\mathrm{a}_{3}$ is equal to the number of ways of selecting 2 elements out of $\mathrm{n}-1$ elements. So, the required number of subsets is ${ }^{\mathrm{n}-1} \mathrm{C}_{2}$</div>


title: the-number-of-coulombs-required-for-the-deposition-of-108-g-of-silver-is
slug: the-number-of-coulombs-required-for-the-deposition-of-108-g-of-silver-is-32465
tags: ['Chemistry', 'Electrochemistry', 'BITSAT', 'BITSAT 2011']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">96500</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">The number of coulombs required for the deposition of $108 \mathrm{~g}$ of silver is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">96500</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">48250</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">193000</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">10000</span> </li> </ul>

answer:
<div class="solution">Amt. deposited $=\frac{\text { E. wt } \times \text { Q }}{96500}$;<br/>$$<br/>107.870=\frac{107.870}{96500} \times \mathrm{Q} ; \quad \therefore \mathrm{Q}=96500 \mathrm{C}<br/>$$</div>


tags: ['Mathematics', 'Permutation Combination', 'BITSAT', 'BITSAT 2022']
source_db: MarksBatch2_P3.db
title:
the-number-of-different-sevendigit-numbers-that-can-be-written-using-only-the-three-digit-1-2-and-3-with-the-condition-that-the-digit-2-occurs-twice-i

slug:
the-number-of-different-sevendigit-numbers-that-can-be-written-using-only-the-three-digit-1-2-and-3-with-the-condition-that-the-digit-2-occurs-twice-i-89069

question:
<div class="question">The number of different seven-digit numbers that can be written using only the three digit 1 , 2 and 3 with the condition that the digit 2 occurs twice in each number is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"> <br/>${ }^7 C_2 2^5$
<br/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"> <br/>${ }^7 p_2 2^5$
<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"> <br/>${ }^7 C_2 5^2$
<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"> <br/>None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"> <br/>${ }^7 C_2 2^5$
<br/></span> </div>

answer:
<div class="solution">Other than 2, remaining five places can be filled by 1 and 3 for each place. The number of ways for five places is $2 \times 2 \times 2 \times 2 \times 2=2^5$. For 2 , selecting 2 places out of 7 is ${ }^7 C_2$. Hence, the required number of ways is ${ }^7 C_2 \times 2^5$.</div>


title: the-number-of-double-bonds-in-gammexane-is-1
slug: the-number-of-double-bonds-in-gammexane-is-1-87906
question: <div class="question">The number of double bonds in gammexane is:</div>
tags: ['Chemistry', 'Hydrocarbons', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">3</span> </li> </ul>

answer:
<div class="solution">Gammexane is $\mathrm{C}_{6} \mathrm{H}_{6} \mathrm{Cl}_{6}$ or (6,6,6) . It is a saturated compound so no double bond is there in it.
<br/> <br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/3tAXFF6g0GSv7lY08sNHVUfOJyUGEyGHLZ6JtVQHrLM.original.fullsize.png"/></div>


title: the-number-of-elements-present-in-the-dblock-of-the-periodic-table-is
slug: the-number-of-elements-present-in-the-dblock-of-the-periodic-table-is-78220
question: <div class="question">The number of elements present in the d-block of the periodic table is</div>
tags: ['Chemistry', 'd and f Block Elements', 'BITSAT', 'BITSAT 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">40</span> </div>
answer: <div class="solution">40 elements are present in d-block.</div>
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">40</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">41</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">45</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">46</span> </li> </ul>


title: the-number-of-enantiomers-of-the-compound-ch-3-chbrchbrcooh-is
slug: the-number-of-enantiomers-of-the-compound-ch-3-chbrchbrcooh-is-98443
tags: ['Chemistry', 'General Organic Chemistry', 'BITSAT', 'BITSAT 2013']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">The number of enantiomers of the compound $\mathrm{CH}_{3} \mathrm{CHBrCHBrCOOH}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">3</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">4</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">The structure $\mathrm{CH}_{3}{ }^{*} \mathrm{CH} \mathrm{Br} \mathrm{C}^{*} \mathrm{H} \mathrm{Br} \mathrm{COOH}$
<br/> <br/>has two different chiral carbon atoms, hence number of enantiomers (optically active forms) is $2^{n}=2^{2}=4$</div>


title: the-number-of-geometrical-isomers-of-ch-3-ch-ch-ch-ch-ch-chcl-is
slug: the-number-of-geometrical-isomers-of-ch-3-ch-ch-ch-ch-ch-chcl-is-30075
tags: ['Chemistry', 'General Organic Chemistry', 'BITSAT', 'BITSAT 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">8</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">The number of geometrical isomers of $\mathrm{CH}_{3}-\mathrm{CH}=\mathrm{CH}-\mathrm{CH}=\mathrm{CH}-\mathrm{CH}=\mathrm{CHCl}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">4</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">6</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">8</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">The given structure has three double bonds whose each carbon atom is differently substituted hence number of geometrical isomers will be $2^{n}=2^{3^{\prime}}=8$, where $n$ is the number of double bonds whose each carbon atom is differently substituted.</div>


tags: ['Mathematics', 'Circle', 'BITSAT', 'BITSAT 2016']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">16</span> </div>
source_db: MarksBatch2_P3.db
title:
the-number-of-integral-values-of-for-which-x-2-y-2-x-1-y-5-0-is-the-equation-of-a-ircle-whose-radius-cannot-exceed-5-is

slug:
the-number-of-integral-values-of-for-which-x-2-y-2-x-1-y-5-0-is-the-equation-of-a-ircle-whose-radius-cannot-exceed-5-is-26885

question:
<div class="question">The number of integral values of $\lambda$ for which $\mathrm{x}^{2}+\mathrm{y}^{2}+\lambda \mathrm{x}+(1-\lambda) \mathrm{y}+5=0$ is the equation of a ircle whose radius cannot exceed $5,$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">14</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">18</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">16</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None</span> </li> </ul>

answer:
<div class="solution">Radius $\leq 5$
<br/> <br/>$$
<br/> <br/>\begin{array}{l}
<br/> <br/>\sqrt{\frac{\lambda^{2}}{4}+\frac{(1-\lambda)^{2}}{4}-5} \leq 5 \Rightarrow \lambda^{2}+(1-\lambda)^{2}-20 \leq 100 \\
<br/> <br/>\Rightarrow 2 \lambda^{2}-2 \lambda-119 \leq 0 \\
<br/> <br/>\therefore \frac{1-\sqrt{239}}{2} \leq \lambda \leq \frac{1+\sqrt{239}}{2} \Rightarrow-7.2 \leq \lambda \leq 8.2 \\
<br/> <br/>\quad(\text { approx. }) \\
<br/> <br/>\therefore \lambda=-7,-6,-5, \ldots \ldots .7,8, \text { in all } 16
<br/> <br/>\end{array}
<br/> <br/>$$ values</div>


title: the-number-of-points-at-which-the-function-f-x-l-o-g-x-1-is-discontinuous-is
slug: the-number-of-points-at-which-the-function-f-x-l-o-g-x-1-is-discontinuous-is-89523
tags: ['Mathematics', 'Continuity and Differentiability', 'BITSAT', 'BITSAT 2016']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">The number of points at which the function $\mathrm{f}(\mathrm{x})=\frac{1}{\log |\mathrm{x}|}$ is discontinuous is:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">3</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">4</span> </li> </ul>

answer:
<div class="solution">The function $\log |x|$ is not defined at $x=0$, so, $\mathrm{x}=0$ is a point of discontinuity. Also for $\mathrm{f}(\mathrm{x})$ to be defined, $\log |\mathrm{x}| \neq 0 \Rightarrow \mathrm{x} \neq\pm 1$
<br/> <br/>Hence, 0,1,-1 are three points of discontinuity.</div>


title: the-number-of-positive-integral-solution-of-abc-30-is
slug: the-number-of-positive-integral-solution-of-abc-30-is-64611
tags: ['Mathematics', 'Permutation Combination', 'BITSAT', 'BITSAT 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">27</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">The number of positive integral solution of $\mathrm{abc}=30 \mathrm{is}$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">30</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">27</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">8</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

answer:
<div class="solution">We have : $30=2 \times 3 \times 5 .$ So, 2 can be assigned to either a or b or ci.e. 2 can be assigned in 3 ways. Similarly, each of 3 and 5 can be assigned in 3 ways. Thus, the number of solutions is $3 \times 3 \times 3=27$</div>


tags: ['Mathematics', 'Permutation Combination', 'BITSAT', 'BITSAT 2020']
source_db: MarksBatch2_P3.db
title:
the-number-of-possible-outcomes-in-a-throw-of-n-ordinary-dice-in-which-at-least-one-of-the-dice-shows-an-odd-number-is

slug:
the-number-of-possible-outcomes-in-a-throw-of-n-ordinary-dice-in-which-at-least-one-of-the-dice-shows-an-odd-number-is-58203

question:
<div class="question">The number of possible outcomes in a throw of n ordinary dice in which at least one of the dice shows an odd number is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$6^{n}-1$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$3^{n}-1$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$6^{n}-3^{n}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$6^{n}-3^{n}$</span> </div>

answer:
<div class="solution">Total number of ways $=6 \times 6 \times \ldots \ldots$ to $n$ times $=6^{n}$ Total number of ways to show only even number $=3 \times 3 \times \ldots \ldots$ to $n$ times $=3^{n}$
<br/> <br/>$\therefore$ required number of ways $=6^{\mathrm{n}}-3^{\mathrm{n}}$</div>


tags: ['Mathematics', 'Trigonometric Equations', 'BITSAT', 'BITSAT 2019']
source_db: MarksBatch2_P3.db
title:
the-number-of-real-numbers-for-which-the-equality-sin-sin-cos-cos-1-holds-for-all-real-which-are-not-integral-multiple-of-2-is

slug:
the-number-of-real-numbers-for-which-the-equality-sin-sin-cos-cos-1-holds-for-all-real-which-are-not-integral-multiple-of-2-is-27135

question:
<div class="question">The number of real numbers <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi></math> for which the equality <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>sin</mi><mo>(</mo><mi>λ</mi><mi>α</mi><mo>)</mo></mrow><mrow><mi>sin</mi><mi>α</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><mi>cos</mi><mo>(</mo><mi>λ</mi><mi>α</mi><mo>)</mo></mrow><mrow><mi>cos</mi><mi>α</mi></mrow></mfrac><mo>=</mo><mi>λ</mi><mo>-</mo><mn>1</mn></math>, holds for all real <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> which are not integral multiple of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>π</mi><mo>/</mo><mn>2</mn></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Infinite</span> </li> </ul>

answer:
<div class="solution"><p>We have, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>sin</mi><mfenced><mrow><mi>λ</mi><mi>α</mi></mrow></mfenced></mrow><mrow><mi>sin</mi><mi>α</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><mi>cos</mi><mfenced><mrow><mi>λ</mi><mi>α</mi></mrow></mfenced></mrow><mrow><mi>cos</mi><mi>α</mi></mrow></mfrac><mo>=</mo><mi>λ</mi><mo>-</mo><mn>1</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mfenced><mrow><mi>λ</mi><mi>α</mi></mrow></mfenced><mi>cos</mi><mi>α</mi><mo>-</mo><mi>cos</mi><mfenced><mrow><mi>λ</mi><mi>α</mi></mrow></mfenced><mi>sin</mi><mi>α</mi><mo>=</mo><mfenced><mrow><mi>λ</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mi>sin</mi><mi>α</mi><mi>cos</mi><mi>α</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mfenced><mrow><mi>λ</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mi>α</mi><mo>=</mo><mfenced><mrow><mi>λ</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mi>sin</mi><mi>α</mi><mi>cos</mi><mi>α</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mfenced><mrow><mi>λ</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mi>α</mi><mo>=</mo><mfrac><mrow><mi>λ</mi><mo>-</mo><mn>1</mn></mrow><mn>2</mn></mfrac><mi>sin</mi><mn>2</mn><mi>α</mi></math></p><p>It is possible, when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>-</mo><mn>1</mn><mo>=</mo><mn>0</mn><mo>⇒</mo><mi>λ</mi><mo>=</mo><mn>1</mn></math></p><p>and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>λ</mi><mo>-</mo><mn>1</mn></mrow><mn>2</mn></mfrac><mo>=</mo><mn>1</mn><mo>⇒</mo><mi>λ</mi><mo>=</mo><mn>3</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mi>λ</mi></math> has only two solution.</p></div>


